What is the origin of the terms Bogart as in Bogart that drink?

Answer 1

Bogarting is generally a term used when referring to smoking, particularly a joint. The actor, Humphrey Bogart was notorious for talking with a cigarette hanging out the side of his mouth. Bogarting a joint implies one who is keeping the joint in his mouth instead of passing it when finished "taking a hit". Due to this idea of hoarding the joint for yourself bogarting has also become a term used to imply one who refuses to share.

In response to the previous answer when Bogart asks Sam in Casablanca to "play it again", he is referring to a song Sam is to play on the piano. The song is a tune which was important to Rick, Bogart's character, and Ilsa, Bergman's character. The fact he is getting a drink at this moment is irrelevant to what he is saying.