A cool English teacher is not always necessarily an English teacher, and not even necessarily a teacher. They are usually an authority figure or mentor of some sort. They are never cool.
Cool English teachers want to set themselves apart from the other teachers. They want to teach the kids about life. They want to make learning relevant. They are okay with informalities and relaxation of rules, and are sometimes willing to look the other way. They may dress up subject matter by trying to connect it with things that are popular with the kids. They will try to use and understand slang and be familiar with youth subculture. They may promote a "subversive" political or social agenda.
As mentioned, they are never cool. An actual cool teacher is never of the "cool English teacher" type. They are cool because they actually pay attention to their students and want to teach them. A "cool English teacher" is either adapting a facile personality on a condescending way, or is in need of validation and acceptance by their students.