Krumple, in principle I think your explanation is correct, and it may well be the one most comprehensible to the OP. I do, however, think statements like these are somewhat inaccurate, technically speaking:
So there is a force acting on the object in the opposite direction of the elevator's travel (downward). So the blood or water get's squished against the floor of the elevator due to this force.
The reason it's called a "pseudo-force" is because there is no force "pushing down on" and "squishing" the balloon. The phenomenon is an artifact of motion (acceleration) rather than the application of a typical physical "force," such the head of a hammer hitting a nail.
Mash down on the gas pedal of your hot rod Lincoln and you will feel as though you are being pushed backwards (but there's nothing pushing you). Hit a tree in that hot rod while going 100 mph, and your head will go through the windshield. In neither case is there any tangible physical force which is pushing/pulling you.
But you're right that it is the existence of the inertial mass that creates this effect. It would not occur without inertia (which mass is said to measure). Even though we have a label for it, it seems no satisfactory answer to the question of the origin of inertia has been found.