The phrase Poretz Geder in the context of being one who violates rules, like in your question, can be understood by looking at the two words which make up the phrase as found in Sefer HaAruch from Rabbi Natan of Rome.
He clarifies the meaning of peretz (פֶּרֶץ) from Brachot 17b which says:
“There is no breach”; that our faction of Sages should not be like the faction of David, from which Ahitophel emerged, who caused a breach in the kingdom of David.
“And no going forth”; that our faction should not be like the faction of Saul, from which Doeg the Edomite emerged, who set forth on an evil path.
“And no outcry”; that our faction should not be like the faction of Elisha, from which Geihazi emerged.
“In our open places”; that we should not have a child or student who overcooks his food in public, i.e., who sins in public and causes others to sin, as in the well-known case of Jesus the Nazarene.
And also from Sukkah 26a which discusses the idea that a breach (פִּרְצָה) summons the thief.
Rava said: A breach summons the thief. If the guardian builds a sukka, thieves will know where the guardian is located in the field and they will enter the field elsewhere. The exemption of the watchman from the mitzva of sukka prevents that situation.
This implies that the poretz not only personally behaves in a deviant fashion in public, but that they also have influence on others to cause them to take bad actions. They summon the thief.
The Aruch defines geder as it is explained from Brachot 17a which says:
מרגלא בפומייהו דרבנן דיבנה אני בריה וחברי בריה אני מלאכתי בעיר והוא מלאכתו בשדה אני משכים למלאכתי והוא משכים למלאכתו כשם שהוא אינו מתגדר במלאכתי כך אני איני מתגדר במלאכתו ושמא תאמר אני מרבה והוא ממעיט שנינו אחד המרבה ואחד הממעיט ובלבד שיכוין לבו לשמים.
He says that geder there has a connotation of authority. The Aruch also says that there are those who say in that context that geder has the connotation of Torah study.
So using these two components, the phrase Poretz Geder means someone who doesn't learn Torah and doesn't acknowledge the authority of the Torah, who violates it publicly and leads others on that path, which is a definition with a negative connotation.
This is in keeping with your specific example from Mishnah Berurah 134:2 which attributes the requirement of saying V'Hu Rachum to a Takanah of the Soferim as found in Kol Bo. This person, who refuses to say it in minyan, meaning in public, which has influence on others, is called a Poretz Geder. In your specific example, it would be alluding to someone who denies the authority of the Oral Torah like is explained in the name of Rava in Eruvin 21b.
א"ל וכי תורה פעמים פעמים ניתנה אלא הללו מדברי תורה והללו מדברי סופרים דרש רבא מאי דכתיב (קהלת יב, יב) ויותר מהמה בני הזהר עשות ספרים הרבה וגו' בני הזהר בדברי סופרים יותר מדברי תורה שדברי תורה יש בהן עשה ולא תעשה ודברי סופרים כל העובר על דברי סופרים חייב מיתה
This also seems to be the understanding and contrast of these words as found in Brachot 63a-b and the discussion of Rabbi Chanina, son of the brother of Rabbi Yehoshua when he left Israel to teach Torah in Bavel.
התחיל הוא מטמא והם מטהרים הוא אוסר והם מתירים הכריז עליהם אנשים הללו של שוא הם של תהו הם אמרו לו כבר בנית ואי אתה יכול לסתור כבר גדרת ואי אתה יכול לפרוץ
The general discussion on this page is talking about when an individual should work to build Torah learning and mitzvah performance doing outreach in their particular community and when they shouldn't. One of the examples brought to illustrate this is from Rabbi Chanina. The complete story in translation is:
Rav Safra said: Rabbi Abbahu would relate: When Ḥanina, son of Rabbi Yehoshua’s brother, went to the Diaspora, Babylonia, he would intercalate years and establish months outside of Eretz Yisrael. Because Judaism in Eretz Yisrael had declined in the wake of the bar Kokheva rebellion, he considered it necessary to cultivate the Jewish community in Babylonia as the center of the Jewish people. Among other things, he intercalated the years and established the months even though the halakha restricts those activities to Eretz Yisrael. Eventually, the Sages of Eretz Yisrael sent two Torah scholars after him, Rabbi Yosei ben Keifar and the grandson of Zekharya ben Kevutal. When Ḥanina saw them, he asked them: Why did you come? They responded: We came to study Torah. Since he saw his standing enhanced by the Sages of Eretz Yisrael coming to study Torah from him, he proclaimed about them: These people are eminent scholars of our generation, and their fathers served in the Temple. As we learned in tractate Yoma: Zekharya ben Kevutal says: Many times I read before the High Priest from the book of Daniel on the eve of Yom Kippur. These two scholars, however, began to dispute every decision Ḥanina rendered in response to questions raised in the study hall. He ruled it impure and they ruled it pure; he prohibited it and they permitted it. Eventually, he proclaimed about them: These people are worthless. They are good for nothing and they know nothing. They said to him: You have already built up our names and glorified us; you cannot now demolish. You have already built a fence and you cannot break through it. He said to them: Why is it that when I rule something impure, you rule it pure; when I prohibit it, you permit it? They said to him: We do this because you intercalate the years and establish the months outside of Eretz Yisrael. He said to them: Didn’t Rabbi Akiva ben Yosef also intercalate years and establish months outside of Eretz Yisrael? They replied to him: Leave the case of Rabbi Akiva, as, when he left, he did not leave behind anyone as great in Torah as he in Eretz Yisrael. Rabbi Ḥanina said to them: I also did not leave behind anyone as great as me in Eretz Yisrael. They said to him: The kids who you left behind have grown into goats with horns; they are greater than you are. And they sent us to you, and this is what they said to us: Go and tell him in our name: If he obeys, fine; and if he does not obey, he will be ostracized.
The idea of building a fence is when Rabbi Chanina lauded the two visitors from Israel as eminent scholars and the idea of breaking that boundary was when he said they were worthless, good for nothing and knew nothing.
It is important to point out that in Torah, there is another usage of this expression which is not negative. It is like is found in Sefer Kehillat Yaacov by Rabbi Yaacov Tzvi Yolles who also links the phrase with the manifestation of G-d's Kingship.
