In summary, from an Irishman's point of view:
"Paddy's Day" is the general, colloquial usage among Irish people. Stress first word.
"Patrick's Day" is very common - a little more stress on second word, usually.
"Saint Patrick's Day" is near-formal, more used in writing.
In Ireland, "Patti's Day" / "Patty's Day" is just asking for trouble!

Regards

;>)>