Standing when bringing in the Shabbat originates from BT (Shab. 119a):
רבי חנינא מיעטף וקאי אפניא דמעלי שבתא אמר בואו ונצא לקראת שבת המלכה רבי ינאי לביש מאניה מעלי שבת ואמר בואי כלה בואי כלה
Trans. (Sefaria):
Rabbi Ḥanina would wrap himself in his garment and stand at nightfall on Shabbat eve, and say: Come and we will go out to greet Shabbat the queen. Rabbi Yannai put on his garment on Shabbat eve and said: Enter, O bride. Enter, O bride.
In Talmudic times there was no official liturgical piece read which signified the onset of Shabbat. Nevertheless, the rabbi(s) deemed proper to invite the Shabbat while standing as a gesture of respect. The practice was codified in the halachah (cf. Tur & SA 262:3). Later on, when much of Jewry had distinct formulas for inviting the Shabbat, we find Be'er Heitev (op. cit. 5) who writes that "we" stand as one would for a respectable person (without mention at what particular point to stand for). Again, later on, Aruch HaShulchan (op. cit. 5) writes:
ועכשיו אנו אומרים לכבודה מזמורים ולכה דודי, ובחרוז האחרון עומדים וחוזרים פנינו כלפי הפתח, ואומרים: 'בואי בשלום... בואי כלה בואי כלה'.
Trans:
And nowadays we say in her [Shabbos] honor various psalms and "Lecha Dodi". For the last verse, we stand and face the [synagogue's] entrance and say "בואי בשלום וכו׳".
Generally speaking, there was no specific point when one must stand when inviting the Shabbat. It appears to have been somewhat similar to a matter of etiquette; stand when inviting the honorable guest. Those who have the custom to stand only for the last verse, בואי בשלום, base the practice on the Talmud (despite the fact that all other liturgy commonly recited prior to that verse was not recited in Talmudic times).