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.