Mass: Matter or Inertia?

energy is the ability to do work, if we let work be PdV instead of Fds you may be pleased. i dont have a problem with circular definitions

Gib Z, you're not alone. The confusion between inertia, mass, matter and momentum is still common today in text books and physics papers.
As kwestion said you are "onto something that is important at a fundamental level."
In quantifying one or the other we often negate or assume a certain relative quantity.
What is important, is to understand the fundamental conceptual differences first, so the math does not become ambiguous.

A body does not move of its own accord. This seems incredibly obvious from classical physics to today.
It essentially says, unless an external force is applied, a body will remain at "rest".
Why? Because in classical terms, inanimate bodies, are inert. (unable to move or act)
So matter is inert, force is the antithesis of matter, the two together are action. (not to be confused with motion)
The idea that a body in motion will remain in motion is just as obvious but is more easily understood as
a relative property of rest. A body at rest remains at rest even it that state of rest is "motion" with respect to another observer. So now a body is both at rest and in motion depending on the frame of reference taken.
But in either case, it still has inertia - it will not change of its own accord.
Now we want to quantify the force required to change the state of motion of this body.
When its at rest, with respect to one observer, no matter how fast it is moving with respect to any other, the force required is always the same. This force is a direct measure of the quantity of matter of the body and since motion does not change the total number of particles in a body, rest mass is constant.
As a measure of energy, that quantity of matter is called mass.
When the same body is measured by an other observer that measures the body to be in motion, the force required to bring it to rest depends on the speed of the body with respect to that observer. This force is called momentum. The momentum is a measure of the bodies mass and speed.
To recap:
Inertia is a property of Matter (the property that it remains inert, regardless its state of motion)
Mass is the constant, quantifiable energy of matter
Momentum is a relative measure of Mass.

Then along came Einstein and everyone started mixing metaphors.
As soon as E=mc^2 appeared everyone began to talk of mass as energy , which is fine as long as nobody
confuses a quantity of energy as a statement of a quantity of matter.
The most common confusion was energy expressed as relativistic mass.
This was analogous to saying: 2+2 = 4 unless you start with 3.
The remaining expressions are much less so, but still prone to confusion.
Gravitational mass, attractive gravitational mass, passive gravitational mass, "energy" of momentum, inertial "energy"...

Mass is a quantitative measure of inertia.

Apparently, a photon imparting momentum to a mass (or vice versa) allows inertia for the mass but not for the photon.

Apparently, a photon imparting momentum to a mass (or vice versa) allows inertia for the mass but not for the photon.
Forgive me, but... what?

If the definition of inertia includes changes in momentum, by its definition how do interactions of photons (non-inertial) or masses (inertial) differ otherwise?

Photons and mass are two different subjects. They both 'obey' the geometric properties of spacetime though. That is, both will 'bend' to gravity. Photons have what's called an instant acceleration, relative or rest mass do not. Otherwise I agree in that it is a 'muddy' question :)

We can slow down light and even stop it, when we do that it disappear, as far as I understand, only to show itself when we accelerate it again. Then again, I might be wrong here. On the other hand, electrons are not defined particles either, they just have a probability focus, right:)

And inertia is a property of relative / rest mass which differs from photons instant acceleration. But I have problems with exchanging mass for inertia as inertia is a intangible property concurring from mass, whereas mass does not, as far as I know, create itself by experience inertia.