St. Patty or St. Paddy?

Happy St. Patricks Day!

I have an Irish friend who is very annoyed with us Americans who say "Happy St. Patty's Day"- she said that it is "Paddy". I'd trust her, but isn't Patty the short version of Patrick? Can anyone settle this?
Update: I'd never call a friend "Patrick", Patty- it's a girl's name here- but if you ever walk around on St. Patrick's Day, there are tons of signs saying "Happy St. Patty's Day!" Thanks to the Irish for clearing this one up!
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