I don’t think you experience zero g, you feel acceleration right?

Having done "zero-g" in aircraft many times, you do feel it. Your blood shifts. Your spine decompresses. Your head gets lighter. You very much know that you are falling.

That's not true zero-g. In a plane you are experiencing the force of acceleration towards Earth, which is closer to 1G than to 0G.

A lot of confusion here probably comes from the fact that g-force is not the same as acceleration. It's apparent acceleration. So 0g in an aircraft is entirely possible. So is 0 acceleration but that's not 0 g, that's 1 g.

When do people mean anything other than "free fall" by "zero g"?

I can assure you that while being trained in basic aerobatics we regularly went well beyond the zero-to-one G range.

  the cat experiences "zero-g" as it accelerates towards terminal velocity

I think you meant the cat experiences 'zero-g' AFTER it's done accelerating and has reached terminal velocity.

I'm guessing in most cat falling situations the cat does not reach terminal velocity, so it is accelerating the whole time and can adjust based on the direction of acceleration.

No. Terminal velocity is one-g, the same as sitting in a chair. A skydiver at terminal velocity feels exactly as much force/acceleration from the wind as they would feel force from lying down on the ground.

That's right, the forces cancel each other out, so that's "zero-g".

If you're accelerating, you're experience some "g" force, not "zero-g".

I’m sitting on the toilet right now, and the forces are canceling each other out so that I’m not sitting in the toilet. Is this 0g?

Except acceleration due to gravity. Freefall is acceleration without feeling any apparent force.