I'm saying yes, however, the reason for this probably has to do with different definitions. The definition I use to describe villains is in a more classical sense as by definition they are evil("a cruelly malicious person who is involved in or devoted to wickedness or crime"-Dictonary.com). I like to use villain in that sense because I think they should be a use for the word antagonist as a more benevolent and/or more down to earth version of a villain. When I think of villains, I don't think of Zuko, I'm more inclined to place him under the umbrella of an antagonist as opposed villains since the word villain has this certain connotation due to the people it is often used to describe.

I still feel really bad for Lots-o
:'(

Freeze is a pretty good example.

Depends on the villain