I'm Liveinjapan.

Can it be called a handle, handle name, handle to my name, or nickname?

nickname, forum name, handle
are all OK, IMO

In some forums with two-level identification the above are different from the login or user names.
I'd say it's your "nickname" here in EF (or just "nick").
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I used to hear the term 'handle' used like that, back when talking on CB radio was popular. So I suppose you might call it that.
I'd just say that it's "my computer name", "my forum name". Or sometimes "my login".
 Marius Hancu's reply was promoted to an answer.
 Kooyeen's reply was promoted to an answer.
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Got it! Thanks, Clive, Marius and Kooyeen.