What if, for example, we placed a ball of lead on top of a sponge? That would depress the sponge, forcing air out of it (no matter which model you believe). I think everyone ought to agree that the weight of the ball on top of the sponge would be slightly less than the weight of the ball plus the weight of the sponge.
There is something to what Scepti's saying, as far as this goes. Typically it'd be thought of as density, Scepti's just saying less dense objects essentially have 'more room' for air/atmospheric pressure to get inside.
So an interesting experiment might be to be able to work out how to differentiate between the two, if that's even possible.
I do like the sound of experiment 2 though. Easily doable, and there are clear predictions. By the model of gravity, air's just been replaced with water so their differences in weight wouldn't change that much. By denpressure, if it's down to atmosphere getting into the object, that's all been removed and they'd have relatively equal volume. If it's done quickly...
Does come down to what Scepti says though. Even if there is something you didn't take into account, if nothing else it helps define denpressure in more detail.