I'm not a teacher, so I'm certainly not the best to say what teachers do. The definitions given above are also what I'd say is correct.
In terms of differences of teaching method - Teaching English as a Foreign Language is much more 'traditional' teaching. Probably the way most people learn a language - starting with basics of grammar, vocabulary etc and progressing that way.
Teaching English as a Second Language, as far as I know, is more about teaching people what they require to know to live in that country. A person who is living in Britain probably cares more about knowing how to ask for help in a supermarket than they do about learning 1st, 2nd, and 3rd conditionals. ESL is more concerned with what a person is required to know in order to live an active and fulfilling life in which language is not a barrier to communication.
You'd do well to wait for a teacher of English to correct me, as I'm not a teacher in the slightest.
EDIT - Also, to address the issue of English perhaps being a third, or fourth language, I've seen 'English as a Second or Other Language' used.