What are the halachik definitions of bittul torah?
I.e. some say that studying secular sciences qualifies as bittul torah while others say that it doesn't. How are they defining the halachik definition of bittul torah differently and on what basis?
The Gemara in Megillah 3a:
מכאן סמכו של בית רבי שמבטלין תלמוד תורה ובאין לשמוע מקרא מגילה
The house of Rebbi was mevatel Torah to come and hear the Megillah Much has been written about why the reading of the Megillah is considered bittul Torah, as the Megillah itself is seemingly a part of Torah.
Some explain that the Mitzvah of reading the Megillah is meant to be an act of listening, not an act of learning. The Beis Ephraim in O.C. #68 (paragraph starting מעתה) explains that it is a qualitatively lower level of learning, and there is such a thing as ביטול תורה באיכות - qualitative bittul Torah. (The Beis Ephraim has further proof to this from Berachos 10b and the Rashba there, among other proofs.) The glosses of the Rashash to that Gemara, based on the Shulchan Aruch, says the same thing. The Pele Yo'etz in Chelek 2, Erech Tehillim, 88b also writes that there is Bittul Torah in qualitatively "lesser" learning. The Chassid Yavetz (15th century) in his commentary to Avos 6:4 writes a similar idea - that the learning of Torah is measured according to the one learning, not the material learned. The Shulchan Aruch HaRav (18th century) in Hilchos Talmud Torah 2:5 writes that anyone who reviews what they have already learned because it is easier (as opposed to in order to remember it) literally is responsible for their own life.
Either way, even a mitzvah (even those who understand the Gemara is about the transit time of going to hear the Megillah will agree that the travel to do a mitzvah is also considered a level of mitzvah and even has the exemption from other mitzvos due to being involved in a current mitzvah) is called bittul Torah and requires justification (See the gemara there, which learns from a kal v'chomer that this particular Bittul Torah is acceptable). It seems that any act of not learning needs justification (I am making the assumption that the threshold of justification is more than "I was having fun;" otherwise the Gemara wouldn't very much need to tell us about reading Megillah being enough of a justification, nor to learn that it is a justification from a kal v'chomer).
Rambam presents bittul Torah not as a bittul assei in the normal sense, but rather as an independent category of "dvar Hashem baza".
This is doesnt seem to be a technical halachic legal entity, but rather a directive to our way of life.
For this reason it is unsurprising that rather than writing that one transgresses this for missing a minute of learning, or even for some other technical violation, Rambam writes (Hil. Talmud Torah 1:13) that one "violates" this by completely disregarding Torah.
כִּי דְבַר ה' בָּזָה זֶה שֶׁלֹּא הִשְׁגִּיחַ עַל דִּבְרֵי תּוֹרָה כָּל עִקָּר.
Regarding the extent of the obligation to study Torah, although Rambam lauds constant involvement with Torah and wisdom, he holds that the obligation is only to learn in the day and in the night; not every minute.
אַף עַל פִּי שֶׁמִּצְוָה לִלְמֹד בַּיּוֹם וּבַלַּיְלָה אֵין אָדָם לָמֵד רֹב חָכְמָתוֹ אֶלָּא בַּלַּיְלָה. לְפִיכָךְ מִי שֶׁרָצָה לִזְכּוֹת בְּכֶתֶר הַתּוֹרָה יִזָּהֵר בְּכָל לֵילוֹתָיו וְלֹא יְאַבֵּד אֲפִלּוּ אֶחָד מֵהֶן בְּשֵׁנָה וַאֲכִילָה וּשְׁתִיָּה וְשִׂיחָה וְכַיּוֹצֵא בָּהֶן אֶלָּא בְּתַלְמוּד תּוֹרָה וְדִבְרֵי חָכְמָה.
The vast majority of Rishonim understand that the Gemara in Nedarim (8a) accords it the view of R. Yochanan in Menachos (99a). According to his view one discharges his obligation to study Torah with the recitation of the shma. Assuming one has done so, according to their view there would be no bittul assei in studying other sciences (as there is in the view of R. Yishmael, R. Yochanan's disputant in Menachos).