I see... so "magic" in your world is synonymous with "miracles", wherease magecraft is the use of mana to cause effects as opposed to more mundane means. Except of course that miracles are dictated by a deity or Fate, and your true magic would be in the hands of mortal magi.

So depending on what the source of our transportation to this world is, it could either be an example of true magic by your definition, a miracle, or a technological cause... which if I'm understanding this right would invalidate one of your true magics as being... well, magic. Correct?