To be honest, I think physics classes carefully articulate things in order to avoid veering into the direction that you're dabbling into. Because things will get complicated.
Let me start with the statement:
pressure stops at solid mass
This is wrong, but it's a very clever attempt at explaining the world. Something changes at the surface of solid matter. However, force is still transmitted.
There is a measurable difference in the properties of a solid block of matter under an atmospheric pressure. You could (for a hypothetical) have a material that changes color when its stressed. We actually have stuff that does very similar things to this. My point is that the surface pressure effects a physically meaningful. A solid is a lattice of atoms that hold their relative positions through electron bonds. A pressure on the solid's surface decreases the distance between the atoms.
Nonetheless, you could remove the pressure entirely and the solid won't fall apart. This is because chemical bonds are roughly a Lennard-Jones potential. This isn't exact, and it differs by the specific bond, but it's more than sufficient for our purposes. This potential is:
This is a "potential" graph for a chemical bond. Here, "potential" is a mathematical concept. It is the anti-derivative of force. Again, I'm speaking fairly specific to solids. But I want you to consider a few things here which are important to get the FULL molecular picture of pressure, which is really what your question asks for:
- A small crystal in normal Earth pressure has atomic positions moved just at little to the left of the very minimum of that graph
- Take that small crystal into space. Now their atomic positions are exactly on the minimum of that graph
- Think of the core of the Earth. Due to high-pressure physics, we believe that may still have some lattice structure. In that case, atomic positions would be to the left of minimum, but far above 0 on the y-axis. This is very cool, because it demonstrates that the pressures there exceed the maximum tensile strength a material can have.
This is what I would use to introduce absolute pressure. Space has zero pressure. Solid matter behaves like a spring. At zero pressure, solid matter is a spring at its resting point.
Now let's talk about relative pressure (sometimes gauge pressure). This is the form of pressure that physics class wants to teach you. That's because Newtonian physics (even much of fluids) can be very well-conceptualized with no mention of atoms whatsoever. When you press your hand on a wall, or your foot on the ground, the force is small compared to the ambient pressure, but ambient pressure permeates everything equally. In those cases, we are dealing with non-isotropic forces - and by that I mean directional.
The definition of pressure actually doesn't care whether it's relative, absolute, or isotropic. It's just force per unit area. That's just a unit, and units reoccur in different contexts all the time.
Basically any force you entertain in basic physics can be re-imagined as a pressure just by taking the cross-section somewhere. This is true for pulling with a rope or pushing with a stick. There's a sign difference between "pulling" and "pushing", and in materials those are experienced as "tensile" and "compressive" non-isotropic stress... which has units of pressure.
I hope that helps, but I won't be surprised if you're just more confused. I should mention the material stress tensor. If you appreciate this, and you appreciate the nature of zero/absolute pressure, then you understand this subject entirely.
In this model, we think of a solid as a box, and we do mathematics on it. You can see there are lots of vectors there. If this is an imaginary box around some gas, all components other than those that push directly toward the center of the box are zero, and that is the idea of an isotropic pressure. If you imagine a solid box floating in space, then all the vectors are zero.
From that standpoint, absolute pressure, and the other kinds of pressure should make sense.