Yep, it’s another edition of “Someone on the Internet is wrong.”
It’s always tempting for people in a subculture to either try to be more like, or less like, the dominant culture. It’s also tempting for them to make sweeping statements. So you get Irish-American Catholics saying that “Catholics don’t do such and such,” when really they mean “Irish-American Catholics, or at least the ones whom I know, don’t do such and such.” Sometimes it’s even tempting to say that “Such and such is against my religion,” when really it just conflicts with one’s own tastes and likes.
The primary giftbringers in Orthodox countries were St. Nicholas of Myra (especially in Russia and Eastern Europe before the Communists), St. Basil of Caesarea (in most parts of Greece and Asia under the rule of the Ottoman Empire), and the Baby Jesus (in various places in Eastern Europe). Russia also used to have the Babushka (a pre-Revolution Russian figure who was similar to La Befana). Today Father Frost has mostly replaced St. Nicholas, but the good saint is making a comeback in other places in his Santa Claus guise.
Here’s a very good article about St. Basil’s Day in Holy-Days and Holidays, a collection of articles edited by Deems. (Although the author doesn’t seem to know that every saint’s deathday is his “natalis” or birthday, according to the Church.) Kids received gifts from St. Basil on the night of Dec. 31, St. Basil’s Eve. St. Basil also granted wishes for the forthcoming year. People made giant vasilopita (St. Basil’s cakes) and gave large hunks away to charity, and to carolers who would sing songs about St. Basil. They also ate a nice dish of “St. Basil’s piglet.” It was also the nameday for anyone named Vasilios or the like. Finally, as told by the mother of the author of In the Neighborhood of Zero, St. Basil would appear at night to teach Greek to little children living in Turkish territory.
(Btw, Holy-Days and Holidays reprints a fair number of articles that are dedicated to proving that American Protestantism is good and everything else is bad; but this article is benign, as most of them are. The latter half of the book is about civic American and Canadian holidays, many of them under older names. Both sections of the book feature poetry excerpts of interest. Another interesting publication for holiday info is Foreign Born, a periodical for immigrant New Yorkers.)