Notice that 1) provides the lower limit, based on bulk characteristics that arise naturally from principles of physics. When solid astrophysical objects become sufficiently massive, their own gravitation becomes important to describing their overall shape. They collapse into spheres, or more generally oblate spheroids (due to their rotation) or triaxial ellipsoids (by tidal deformation). In all these cases, hydrostatic equilibrium is fundamental to describing the lowest order of relief. For a good primer on this topic, I can recommend Planetary Surface Processes by H. Jay Melosh, and there is also very nice derivation and discussion of "the potato radius" available here.

2) acts as an upper limit, separating the largest gas giants from the smallest stars or brown dwarfs. The fundamental distinction is whether there are any appreciable nuclear fusion processes occurring, which is largely determined by the mass.

3) is a choice of setting. Classically, planets are "wandering stars". They "wander" from night to night relative to the background stars, because they are close to Earth relative to the stars, a result of sharing a common origin and being bound to the same star. This is where the IAU definition holds only within our solar system. I prefer to generalize to apply in any star system. I.e. a planet does not have to orbit "our" Sun, it can orbit any sun, or system of suns. We can call it an exoplanet if we want to specify that it is outside of our own solar system.

It is important to note that all planets do not necessarily remain bound to their stars. In this case they are called rogue planets, freely wandering through interstellar space. There are thought to be many of these out there.

Another way that 3) might fail is if the object orbits the star indirectly, in which case it may be a moon, or a component of a binary planet, depending on the mass ratio.

Finally, 4) is a statement regarding dynamics. There are a huge number of objects in our solar system that satisfy conditions 1-3. We can, if we like, choose to call them all planets, or we can try to narrow it down. Personally, I prefer having on the order of 10 bodies per system be known as planets, not hundreds or thousands. The question is, can I make and justify that choice through something that is not arbitrary? Yes, I can. I notice that there is a small number of bodies in our solar system that satisfy the first three conditions, yet are also very unique from all the others. I also notice that there is an extremely powerful dynamical process that naturally causes this to happen. I choose to make use of this fact to help me to define what I call a planet.

This is made empirical by such measures as the planetary discriminant, or Stern-Levison Parameter. We see that in our own solar system, there are distinct populations of bodies that are separated by several orders of magnitude through these measures. This means there is something enormously important going on, a physical process that causes a small number of objects to grow to dominate their region of space. That's a result of how planetary accretion and the dynamical evolution of systems work. It naturally separates what we call "planets" from all the other stuff we may call "dwarf planets", "asteroids", etc.