I've only read the first and last bit of the thread (it's just too long), so I apologise if I'm repeating something already posted; I'm posting here because I imagine quite a few people who read this might have done the same.
Father Christmas, St Nicholas and Santa Claus are really three different people. The only thing missing from @Fakeflowersaremynewnormal 's excellent overview is why the 19th century New Yorkers had such a big thing about Sinterklaas.
I guess most people know that New York had once been called New Amsterdam (or Nieuw Amsterdam or something maybe?). The [old] Amsterdammers had always given gifts to each other on 6 December (St Nicholas's Day) because St Nicholas was patron saint of Amsterdam (and of sailors btw).
In the 18th century in the states, resentment towards the British army, with their dedication to St George, moved some New Yorkers decided to form a society of St Nicholas, basically to annoy the British. One member was a mad rich man called John Pintard who had a massive thing for St Nicholas; he founded the New York Historical Society with St Nicholas as its patron. Washington Irving satirised the NY Historical Society by writing a spoof history of New York, poking fun at its romanticism by harping on about the city's supposed Dutch history and St Nicholas's supposed role in the city's development - all just satire of course - published on St Nicholas's day in 1809. It was a best-seller on both sides of the Atlantic. Irving was the originator of Christmas stockings, the flying sleigh, coming down chimneys secretly one night a year to give gifts - all just meant to make fun of Pintard. But everyone loved it, and took it seriously. Then Clement Clarke Moore wrote 'The Night Before Christmas' in 1823 and the gift-giving was transferred from 6 December to Christmas Eve.
Voila Santa Claus, who is, in all the characteristics we most associate with him, entirely American, and mostly the fruit of Washington Irving's imagination, with help from Pintard and Moore. There are lots of celebrations of St Nicholas around the 'Old World' - he was always a very popular saint - but the 'source' St Ncholas is a heretic-bashing bishop from 3rd-century Turkey, so he is portrayed differently in different places around Europe.
The traditional English gift-giving day was New Year's Day, not Christmas Day. The old English Father Christmas didn't give gifts, and he wasn't associated with children, because he was nothing to do with St Nicholas (who is patron saint of both those things). He was just a personification of the spirit of that time of year, like Jack Frost maybe. He was merged with Santa Claus in England during the late 19th century as FakeFlowers says, and was soon indistinguishable from Santa Claus, except the English still feel vaguely like they should use his old name, because the origin of Santa Claus is colonial brit-baiting.
There's a very good website about all of this here - www.stnicholascenter.org/who-is-st-nicholas/origin-of-santa
Sorry for being long-winded!