i.e. are they considered an apikores etc? Or that they just hold an incorrect belief. Previously I had assumed their status was dependent on the principle they denied, so a person who denied the first 5 would be a min, 6 & 7 would be an apikores etc. However after reading this Is it heretical to not accept the writings of the Prophets (besides of Moses)? and the responses it seems apparent that a person who denies the 6th principle is not inherently an apikores.
To answer the question straight-on, Rambam writes regarding all thirteen of his principles:
When a person doubts [any] foundation among these foundations [i.e. his thirteen], he has left the community, denied the principle [i.e. God], and is called a min and a epikores and a cutter of shoots, and it is an obligation to hate him, and of him it is said "shall I not hate those who hate You, God"
Thus, although in Hil. Teshuva (Ch. 3) Rambam differentiates terminologically between those who deny various principles, the end result is the same.
It should be noted, however, that the reason why such a person ends up denying any of the principles might play a role in the application of this severe consequence. See here regarding someone who disbelieves but not out of rebellion (which is not exactly the question asked, but still), and here regarding disbelieving Jews who nevertheless live religiously committed lives, and here for one example of someone who is simply mistaken as to what the principles entail (an issue which is discussed more generally here, in the name of R. Chaim of Brisk).
For reference, the Rambam’s list of 13 Ikkarim is:
- Belief in a G-d who created everything
- Belief in exactly one G-d who is unique like nothing else is
- Belief that G-d does not have a physical form
- Belief that G-d is first and last
- Belief that there is nothing else worth praying to
- Belief in prophecy
- Belief that Moshe Rabbeinu’s prophecy was qualitatively greater than any other prophet who ever lived
- Belief that the Torah we have today is the same Torah Moshe received on Har Sinai
- Belief that the Torah will never be changed
- Belief that G-d knows what goes on in this world
- Belief that G-d rewards good doers and punished evildoers
- Belief in the coming of Mashiach, in spite of his not coming for so long
- Belief in the future resurrection of the dead
The Rambam himself actually lists 24 people who lose their share in the World to Come in Hilchos Teshuvah chapter 3. The list there is:
- One who says there is no G-d who directs Creation
- One who says there are multiple gods
- One who says G-d takes a physical form
- One who says G-d did not create the world
- One who bows to an idol to separate between G-d and man
- One who denies the concept of prophecy
- One who contradicts the prophecy of Moshe Rabbeinu
- One who says G-d doesn’t know what happens on Earth
- One who says G-d did not give even a single passuk or a single word from the Torah, but rather Moshe wrote it on his own
- One who contradicts Torah SheBa’al Peh
- One who says that the Torah was changed, even a single Mitzvah of it
- Deniers of the Resurrection
- Deniers of Mashiach
- One who intentionally violates a particular mitzvah publicly and regularly in order to anger G-d
- One who assimilates during a time of persecution, saying that why should he remain Jewish when this people is being persecuted for doing so? Let me join the nations since they are the stronger party.
- One who intentionally causes the public to sin, regardless of the severity of the sin or the manner in which he does so
- One who separates from communal customs, who does not partake in their sorrows and fasts, but rather goes on his own way as if he was completely separate from them
- One who does Aveiros publicly and insolently, regardless of the severity of the sin
- One who hands over a fellow Jew to the non-Jewish authorities to be killed or attacked
- One who hands over a fellow Jew’s money to the non-Jewish authorities
- Those who put fear on the public not for the sake of Heaven
- Speakers of Loshon Hara
- One who “draws out” their orlah
A simple comparison of the two lists indicates that principle 11 does not merit losing one’s position in the World to Come, while several actions on top of this list of beliefs could also merit losing one’s World to Come. [As Alex pointed out in the comments, while the text in Ani Maamin would seem to place principle 6 on this list as well, the actual text in the Rambam indicates that it’s the same as #6 in Hilchos Teshuvah.]
As noted in the comments to the OP, this list is specifically according to the Rambam’s opinion. Others vary; the Raavad, for one, argues on several of these right there in Hilchos Teshuvah.
Your question is based on a mistaken premise. Nowhere in the linked question, or answers thereto, does it say that one is not an apikorus, min, kofer, heretic, etc. for denying one of Rambam's principles.
The first answer there argues that according to a specific rabbinic authority, one is not a heretic for denying something that is similar to one of Rambam's principles. It does not follow from there that one would not be a heretic for denying something that is actually one of Rambam's principles, nor does it follow that any other rabbinic authorities agree.
(This is not to say that it is necessarily true that one who denies one of Rambam's principles is a min/apikorus/kofer/heretic; rather, it is to say that there is no basis from the linked question to say that it is not true.)
First, there is debate over what the list of iqarei emunah are. The Seifar haIqarim only uses the word "iqar" for postulates, and therefore he only has 3. But he uses the word "shoresh" (literally: root) for mandatory beliefs that are derived by the iqarim. Other derived beliefs which are not mandatory are called "anafim" (branches). Altogether, his iqarim and shorashim cover roughly the same territory as the Rambam's 13, with two real differences:
1- According to the Iqarim, belief in mashiach (the Rambam’s 12th iqar) is an anaf, a branch on the Tree of Life, but not necessary for its survival. So, the Rambam declares a person who doesn’t believe in mashiach a heretic and has no place in the World to Come (Teshuvah 3:6), the Iqarim does not.
2- R’ Albo’s fourth shoresh from his first iqar is that Hashem is uniquely perfect. The Iqarim does include the worthiness of Hashem as a focus of worship as part of His uniqueness. I can not tell is this is part of the shoresh, or an anaf of it. (Which would be prohibited, but as idolatry, not heresy.)
I found a third list of the essentials of faith The mishnah in Avos (3:15 or in some editions 3:11) gives a different list of people who have no portion in the World to Come.
Rabbi Elazar haModa’i (from Modi’in) said, “Someone who desecrates sacred objects, or who disgraces the festivals, or who pales the face of his peer [by embarrassing him] in public, or who annuls the covenant of our father Avraham a”h, or one who interprets the Torah not according to halakhah — even if he has Torah and good deeds, he has no portion in the World to Come.
The Tif’eres Yisrael (ad loc) explains that each action is demonstrative of a lack of belief in a critical belief. He uses the term apiqoreis to refer to such unbelievers.
The first is someone who denies the existence of G-d. He has no reason to acknowledge sanctity.
The second believes in G-d, but believes that the world is eternal. That reality emanates from an impersonal deity and therefore is co-eternal with him. (In short, Platonism.) Such a person denies both creation and G-d’s “Hand” in history, and therefore Shabbos and the holidays are meaningless to him.
The third heretic believes in G-d, who created the world and runs it, but denies the human soul. He believes that the mind is merely the mechanics of the brain and people are thus not different in kind to animals. He has no reason to value human dignity, and therefore nothing stands in the way of his embarrassing others.
The fourth believes in souls, but not the convenant with Avraham. An attitude represented by the one who tries to alter himself to hide the beris milah of that covenant.
The last category is the person who believe in all of the above, but not that the covenant includes the Oral Torah. Therefore, like a Fundamentalist or a Qaraite, he would be lead to concluding that any of his own conclusions drawn from the text are as valid as any other, with no mesoretic process relaying proper and improper derivations.
Still, I have no proof but I think there is consensus. When a beis din for conversion confirms the beliefs of a candidate, they speak in terms of the 13 Iqarim. And when a hekhsher wants to decide who can handle wine in a winery (wine handled by a heretic [apiqoreis, min and/or kofeir, depending on which beliefs are denies] would be stam yeinam), they also ask about the 13 iqarim.
But our 13 iqarim aren't those of the Rambam. The way the Rambam understood his iqarim, anyone who believes in Qabbalah and 10 Sefiros would be a heretic. And the traditional Bal'adi Teimanim reject Qabbalah for this reason. (Bal'adi means national, in contrast to the Shamma'i Teimanim who were influenced by Jews from Shama -- the Levant, mainly Syria.)
I would say observationally, no proof, there is consensus that some loose version of the 13 Iqarim, like those that made it into our siddur as Yigdal and Ani Maamin are part of the halachic definition of who we must treat according to the laws of a heretic. After all, these are the forms that we've been repeating at the start and end of various services now for centuries. They obviously passed rabbinic peer review.
But only part of the definition. Going from heresy to heretic, from belief to person, there is debate as to whether everyone who believes in heresy is subject to the halakhos of how to relate to heretics.
The Radvaz (R David ben Zimra, who was exiled from Spain in 1492 teshuvah #96) rules that someone who honestly studies the sources and reaches a heretical conclusion is not a heretic. And so some rule that all the more so would be someone not raised with Torah, the proverbial tinoq shenishba who therefore believes in heresy, not be a heretic.
The Raavad (Teshuvah 3:7) famously defends a rabbi who he considered far holier than the Rambam even though this rabbi believed that G-d had a body. The Iqarim understands the Raavad to be using the Radvaz's reasoning. The rabbi is holier even though he believes in heresy because he made that mistake though a holy search for truth.
Today this question is most often manifest in the pragmatic issue of whether non-believing Jews can be counted toward a minyan. R' Aharon Soloveitchik ruled that any Jew who believes in a G-d who "Listens" to prayers can be counted toward a minyan. To him, the issue isn't their belief in heresy -- he is following the Radvaz. Instead the issue is whether a Jew for whom prayer lacks its traditional meaning can be counted toward a quorum for praying.
Chabad routinely counts non-believers toward a minyan on the grounds that no one today can be considered at fault for not believing. This leniency is also given by the Chazon Ish, at least in terms of dismissing the practical application of the law of not saving a heretic who is in danger.
Rav Moshe Feinstein does not allows counting non-believing Jews toward a minyan, even if the disbelief is due to upbringing. (Igeros Moshe OC 1:23, 3:12) For Rav Moshe this actually becomes a plus, because that means they are invalid for witnessing too, and any wedding they were the appointed witnesses for is not kosher. Given the number of non-Orthodox women who remarry without the benefit of a gett from their first husband, invalidating the first marriage this way prevents children of the 2nd marriage from being mamzeirim.
In a teshuvah whose preface says its only theoretical, the Binyan Tzion (Rabbi Ettlinger1798-1871, Germany; teshuvah 2:23) rules that we may drink the wine of someone who believes heresy because of upbringing. The Minchas Elazar (the Muncaczer Rebbe; teshuvah #74) insists the preface was added by a publisher, and the responsum was meant to be put into practice. (Rabbi Yona Reiss, of the cRc Beis Din in Chicago, believes the introductory line was added by his son and referred to #23 through the end of vol II.) The Minchas Elazar believes that one can only count a non-believer toward a minyan if they had never before encountered Orthodox Jew. If they know what Orthodox Judaism is, they do not qualify as ignorant by upbringing, as tinoqos shenishba.
So, I would conclude that in practice, we define heresy as a loose version of the 13 iqarim. But whether that is sufficient to make someone a heretic, or if only someone who embraces heresy in an act of rebellion has the halakhos of a heretic (i.e. of an apiqoreis, min and/or kofeir), is an open dispute.