I usually use the expression "see you" to indicate something like "until next time", but the expression "see you" actually contains a verb "see" which should imply, at least that's why I think, that I'm currently seeing the person, since "see you" could be a shorthand for "see you around", "see you next time", even though these are just guesses.
Is it correct to say "see you" when not really seeing the person, for example when you're leaving a chat or writing an e-mail? Would there be a more appropriate expression for these situations?