The place of one's residence is irrelevant in at least one example brought in Shabbos 92a-b:
אָמַר רַב מִשּׁוּם רַבִּי חִיָּיא: הַמּוֹצִיא מַשּׂאוֹי בְּשַׁבָּת עַל רֹאשׁוֹ — חַיָּיב חַטָּאת, שֶׁכֵּן אַנְשֵׁי הוּצָל עוֹשִׂין כֵּן. וְאַנְשֵׁי הוּצָל רוּבָּא דְעָלְמָא?! אֶלָּא, אִי אִיתְּמַר הָכִי אִיתְּמַר: אָמַר רַב מִשּׁוּם רַבִּי חִיָּיא: אֶחָד מִבְּנֵי הוּצָל שֶׁהוֹצִיא מַשּׂוֹי עַל רֹאשׁוֹ בְּשַׁבָּת — חַיָּיב, שֶׁכֵּן בְּנֵי עִירוֹ עוֹשִׂין כֵּן. וְתִיבְטַל דַּעְתּוֹ אֵצֶל כׇּל אָדָם! אֶלָּא, אִי אִיתְּמַר הָכִי אִיתְּמַר: הַמּוֹצִיא מַשּׂוֹי עַל רֹאשׁוֹ — פָּטוּר, וְאִם תִּמְצָא לוֹמַר אַנְשֵׁי הוּצָל עוֹשִׂין כֵּן — בָּטְלָה דַּעְתָּן אֵצֶל כׇּל אָדָם.
Rav said in the name of Rabbi Ḥiyya: One who carries out a burden on his head on Shabbat is liable to bring a sin-offering, as the people of Hotzal do so. They would typically carry burdens on their heads. The Gemara asks: And do the people of Hotzal constitute the majority of the world? Even if in one place it is a typical method of carrying a burden, it remains an atypical method of carrying in the rest of the world. Rather, if this ruling was stated, it was stated as follows. Rav said in the name of Rabbi Ḥiyya: If a resident of Hotzal carried out a burden on his head on Shabbat he is liable, as the people of his city do so. The Gemara asks again: Even if the inhabitants of his city do this, let his intention be rendered irrelevant by the opinions of all other people. If an individual or small group of people conduct themselves in an atypical manner, their conduct is not rendered typical. Typical conduct is determined by the majority of people. Rather, if this was stated, it was stated as follows. One who carries out a burden on his head is exempt. And if you say that the people of Hotzal do so and therefore they should be liable, their intention is rendered irrelevant by the opinions of all other people.
In answer to the main question, "What constitutes a shinui?," the Mishnah there says
הַמּוֹצִיא, בֵּין בִּימִינוֹ בֵּין בִּשְׂמֹאלוֹ, בְּתוֹךְ חֵיקוֹ אוֹ עַל כְּתֵיפָיו — חַיָּיב, שֶׁכֵּן מַשָּׂא בְּנֵי קְהָת. כִּלְאַחַר יָדוֹ, בְּרַגְלוֹ, בְּפִיו וּבְמַרְפְּקוֹ, בְּאׇזְנוֹ וּבִשְׂעָרוֹ, וּבְפוּנְדָּתוֹ וּפִיהָ לְמַטָּה, בֵּין פּוּנְדָּתוֹ לַחֲלוּקוֹ, וּבִשְׂפַת חֲלוּקוֹ, בְּמִנְעָלוֹ, בְּסַנְדָּלוֹ — פָּטוּר, שֶׁלֹּא הוֹצִיא כְּדֶרֶךְ הַמּוֹצִיאִין.
One who carries out an object into the public domain on Shabbat, whether he carried it out in his right hand or in his left hand, whether he carried it in his lap or on his shoulders, he is liable. All of these are typical methods of carrying out an object, as this was the method of carrying the sacred vessels of the Tabernacle employed by the sons of Kehat in the desert. All labors prohibited on Shabbat are derived from the Tabernacle, including the prohibited labor of carrying out from domain to domain. But one who carries an object out in an unusual, backhanded manner, or with his foot, or with his mouth, or with his elbow, with his ear, or with his hair, or with his belt [punda] whose opening faced downward, or between his belt and his cloak, or with the hem of his cloak, or with his shoe, or with his sandal, he is exempt because he did not carry it out in a manner typical of those who carry.
To focus on the last sentence, whether a shinui is present depends on whether the manner of the action is typical or not, for those performing a melacha.
Tosfos there points to Kerisos 13b where Rabbi Meir says one is liable for carrying food in his mouth. Tosfos distinguishes this from carrying other items in the mouth as in the Mishna in Shabbos, since walking while eating is a normal way to carry food. I have likewise been taught not to exit the eruv while eating.
Also, although for carrying there is no distinction between the right and the left, for other melachos this is not always the case. In the melachah of writing there clearly is a distinction; carrying is normal in either hand, but writing is not. A lefty is not halachically in the minority in this regard either, or else he would have an automatic shinui. In fact, in Shabbos 103a, it says regarding a lefty writing:
וְתֶהֱוֵי שְׂמֹאל דִּידֵיהּ כְּיָמִין דְּכוּלֵּי עָלְמָא
Let his left have the status of the whole world's right!
This statement, as it applies to lefties, is accepted as a valid point and is never rebutted. The discussion clarifies that a person could be liable for writing with either hand [only] if he is ambidextrous. Since a lefty's left has the status of the whole world's right, he is part of the majority who write with their dominant hand (whichever that is), and so writing with his left does not constitute a shinui for him.