I suppose that if you just say "Claus" you might get the odd double-take as people think you're talking about something on a cat. "Santa" on its own nearly always means Father Christmas and has the convenience of not causing confusion. In Italy there are hundreds of churches called Santa-something, and there's Santa Lucia of course (which I once sang in a duet with a boatman just off Siracusa in Sicily), but the default understanding, in an appropriate context, of "Santa" is that you're talking about the Christmas chap.

Leonardo added "da Vinci" to his name himself in order to distinguish himself from other local Leonardos. In almost every case if you're talking about "Leonardo" (and, as I keep saying, context is everything), there's no need to add "da Vinci," though you can if you like. "Leonardo da Vinci" is unobjectionable, but referring to a chap called "da Vinci" without the Leonardo is just ignorant and wrong. Incidentally, Doug, it's only "Da Vinci" if you're starting a sentence with it, otherwise it's "da Vinci," never "Da Vinci," and fusing the two bits into a single word is just laughable.

Everybody knows who you mean if you say Michelangelo, and if you use his surname at all (it's Buonarroti plus a few other bits if you're a purist), you're just showing off. In other cases of less-eminent people, you might have to add a helpful bit on to the end on its first use, e.g. "Geoffrey of Monmouth." You wouldn't then go on to calling him just "of Monmouth," would you?