R22 I believe that definition is nowhere to be found, but that the answer to all your questions is yes.

The MM lists "monster types" which are also found on PCs and NPCs. Those are aberration, beast, celestial, construct (including "creatures native to the outer plane of Mechanus"), dragon, elemental, fey, fiend, giant, humanoid, monstrosity, ooze, plant (defined as "vegetable creatures, not ordinary flora"), and undead.

Anything that has such a type has got to count as a creature.