None of those are “burn” decks. They are almost all combo, OTK, or stall/control decks.
Being able to finish from hand does not automatically mean burn deck.
Mozaki was a spell damage OTK deck.
Freeze mage was a stall deck.
I could go on…
What mage has right now is a burn deck. Plays fast to face, curves, and then abandons board, and finishes from hand, with no combos required.
It bears no resemblance to practically any of the decks you named.
Your definition of a burn deck is far too broad imho.