The underlying halacha, as set out in Shulchan Arukh, Orach Chayim 496:3 and elaborated by Arukh HaShulchan, Orach Chaim 496:4-5, makes the governing question not geography alone but intent: a person who reaches a settled place without da'ato lachazor (intent to return) becomes like the locals, while one who still intends to return retains his home community's practice; this hinges on the Gemara's rule that one traveling between places takes on the stringencies of each (Pesachim 51b:5-6), and on the distinct historical basis for the two-day practice described in Mishneh Torah, Sanctification of the New Month 5:11 and Mishneh Torah, Sanctification of the New Month 5:12.
Gray Matter I, Laws of Holidays frames the practical test precisely as da'ato lachazor, citing the Radbaz's rule that bringing or leaving one's family behind is the indicator of whether one intends to return, a standard followed by the Mishnah Berurah; this is the lens through which the NY family's case must be analyzed, since they intend to return home after Yom Tov.
On the specific practical question, the sources diverge. Kitzur Shulchan Arukh Yalkut Yosef, Orach Chaim 496:23, reflecting Rav Ovadia Yosef, rules that bnei chutz la'aretz who go up to Eretz Yisrael intending to return to chutz la'aretz are treated as bnei chutz la'aretz in every respect and must observe the second day of Yom Tov, forbidden in melacha and obligated in the full set of Yom Tov practices, and reports that this is in fact the daily practice among olei regel. By contrast, Igros Moshe OC 3:72 holds that a ben chutz la'aretz who happens to be in Eretz Yisrael for Yom Tov need only observe one day, since the calendar is fixed and known, while the second day remains a practice retained specifically in the Diaspora itself.
Be'er Moshe, Dinei Bnei Eretz Yisrael, Orach Chaim 496 holds that visitors who arrive at a settled community in Eretz Yisrael where one day is observed likewise keep only one day, provided they do not intend to return abroad — which sharpens the dispute, since it ties the ruling back to the very da'ato lachazor criterion that the NY family's plan to go home would seem to implicate.
Igros Kodesh letter 494, addressing the Lubavitcher Rebbe's position on two-day observers, concludes that the second day is kept not because of calendrical doubt but because of the obligation to preserve ancestral custom, and that one should not add even a third day on this basis — a reasoning that supports treating bnei chutz la'aretz as retaining their ancestral two-day practice rather than switching to one day merely by virtue of presence in Eretz Yisrael.
Shut HaRav HaRashi 36:12, from Rav Mordechai Eliyahu, discusses a concrete case of a family that keeps only one day because they are Israelis intending to return to Israel, contrasted with another family who are not strict about the second day, showing how the da'ato lachazor analysis plays out practically in mixed-status households.
Responsa Benei Banim, Volume III 3:4 extends the same da'ato lachazor framework to a bnei chutz la'aretz student living in Eretz Yisrael, ruling, following the Arukh HaShulchan, that one who plans to remain for a year is considered not intending to return and therefore observes only one day — underscoring that the controlling variable throughout this literature is the duration and character of the intended stay, not mere physical location.
Related applications of the same principle appear in Kitzur Shulchan Arukh Yalkut Yosef, Orach Chaim 668:42, which addresses a chutz la'aretz visitor's practice on Shemini Atzeret when staying with hosts in Israel, and in Kitzur Shulchan Arukh Yalkut Yosef, Orach Chaim 668:45, which addresses a bnei Eretz Yisrael resident's conduct on Simchat Torah while temporarily in chutz la'aretz, both reinforcing that the halacha tracks one's residence status and intent rather than mere physical presence in a given place at a given moment.
The broader stringency of the second day once adopted is affirmed in Tur, Orach Chayim 496 and Beit Yosef, Orach Chayim 496:1, both of which describe that whatever is forbidden on the first day is forbidden on the second and that violators are rebuked, while Mishnah Berurah 496:1 explains that the second day is retained even though the calendar is now fixed, out of concern that the practice transmitted from the generations of calendrical uncertainty not be abandoned; Responsa Chatam Sofer, Orach Chayim 145:34 likewise treats the seriousness of maintaining the second day and the reasoning behind its retention even once the date is universally known.
The geographic basis for who is obligated in the two-day practice in the first place is set out in Mishneh Torah, Sanctification of the New Month 3:12, tying the historical origin to where the new-moon messengers could or could not reach, a background relevant to understanding why Eretz Yisrael and chutz la'aretz diverge in the baseline practice reflected also in Beitzah 4b:6's discussion of the two days of the Diaspora as a unified institution.
Peninei Halakhah, Festivals 9:4 addresses the topic of where one must observe the two days of Yom Tov Sheni shel Galuyot directly.