> Which is a consequence of the gravitational pressure, no?Not just gravity alone, no. (And "gravitational pressure" is not a good description of gravity's effect.) Gravity plus the fact that the object does not have a heat source inside it (fusion) to provide thermal pressure. With thermal pressure an object with the same number of particles in it would be much larger--an ordinary star--and its average density would be much smaller.
> the definition of the Schwarzschild radius does not require a black hole
But if the object is not a black hole the Schwarzschild radius is physically meaningless. It only has physical meaning for a black hole, and its physical meaning, as I said, has to do with the area of the hole's horizon, and only that.
> this defines a volume of space, and we call an object whose radius is smaller than its Schwarzschild radius a black hole
No, this is not correct. We call an object a black hole if it has an event horizon. A black hole doesn't have a well-defined "radius" in space any more than it has a well-defined volume. A black hole is certainly not just an object "made of stuff", like a neutron star, that just happens to have collapsed further. It is a different kind of thing from any ordinary object.
> I understand that black holes are fundamentally dynamic processes, but can we not define the behavior at the limit?
What you are describing is not a "limit" of anything. It's simply a wrong description of the physics of a black hole. If by "dynamic" you are talking about the process of forming a black hole from the collapse of a massive object, you're still not correctly describing the end point of that process.