In special relativity is mass just a measure of all other energy than kinetic? The energy momentum equation in special relativity is:
$$E^2=(pc)^2+(mc^2)^2.$$
and it holds for a moving but not accelerating object.
One special case is the massless photon:
$$E=pc.$$
And another one is a resting object:
$$E=mc^2.$$
The first term in the energy momentum equation seems to be kinetic energy of the object as a whole?
That should mean that the second term is all other forms of energy?
This could include kinetic energy of the constituents of the object.
So a full battery for instance has more energy than an empty battery and therefore has a larger mass?
Likewise a hot kilogram prototype has more energy than a cold one and therefore has larger mass? This thermic energy is kinetic energy at constituent level but not for the object as a whole.
In classical mechanics mass is a property of an object that has to do with its inertia. We could define $m=\frac{F}{a}$.
In classical mechanics an empty battery has the same mass as a full one and a cold kilogram prototype has the same mass as a hot one.
Does this not mean that the concept of mass has changed in special relativity and now is a measure of all energy except kinetic energy of an object?
If so $E=mc^2$ does not necessarily predict the atomic bomb?
Instead of it saying that all objects have some intrinsic energy due to their mass it could say that an absolutely cold still object has no mass since it has no energy. Which sounds absurd.
So I am guessing that the old concept of mass from classical mechanics is an approximation to the new concept of mass in special relativity?
 A: Yes, in special relativity, the mass of a system is synonymous with the energy of the system in a frame where its momentum is zero. This, as you observe, would directly follow from the relation $E^2=p^2+m^2$. I will drop the factors of $c$ for convenience (or, in other words, I will use natural units and set $c=1$). Thus, in spirit, saying that the mass is a measure of all energy except the kinetic energy is correct with a couple of caveats: 


*

*As you already notice in the updated version of your question, a many-particle system can have motion of its constituents which do not contribute to the overall momentum of the system but do contribute to the overall energy. And thus, they contribute to the mass of the system as a whole. Thus, the mass of the system as a whole does include contributions from kinetic energy but only from the kinetic energy that doesn't contribute to the overall momentum of the system. 

*Due to the quadratic nature of the relation $E^2=p^2+m^2$, it is a bit problematic to directly identify $p^2$ with the kinetic energy of the system. Rather, the kinetic energy would be $\sqrt{p^2+m^2}-m$ which can be approximated to be $\frac{p^2}{2m}$ as usual for values of $p$ that are very small as compared to $m$. If you naively identify the $p^2$ as the kinetic energy, you wouldn't recover the correct non-relativistic limit. 


Now, all your claims such as a hot cup of coffee having more mass than an otherwise identical but cold cup of coffee are true. However, this doesn't mean that the notion of mass doesn't anymore correspond to the property of inertia. Relativity doesn't change the notion of mass completely--it rather corrects it while unifying it with the notion of energy. In particular, it would be more difficult to accelerate a hot cup of coffee than a cold one if you can measure all the minuscule effects. So, the notion of mass in relativity is yet very much representative of the quality of inertia. The way to see this is to write down the expressions for momentum and energy in relativity. As you probably know, in relativity, $$p=\frac{mv}{\sqrt{1-v^2}}$$$$E=\frac{m}{\sqrt{1-v^2}}$$As you can see, it is the same $m$ that enters the formula for the energy  also enters the formula for momentum. Thus, the same $m$ that represents both the measure of the energy in the rest frame (via entering the formula for energy) and the property of inertia (via entering the formula for momentum). This is the most basic conceptual unification represented in relativity and the genius of Einstein--the (rest) energy of a system is not independent of its inertia but two are the very same thing. 
Now, finally, all of this doesn't mean that a cold object at rest shouldn't have any energy at all. It can very well have all sorts of reasons to have rest energy (and thus, mass). For example, even if all the constituents of a system are at rest and there is no interaction potential energy among them, the system as a whole would still have mass but it would simply be the sum of the masses of all its constituents. So, an object whose constituents are at rest simply means that its mass would not have contributions from the kinetic energy of its constituents. More importantly, there cannot be a massless system with no momentum (i.e., a system known to be rest must have mass). If something has no mass then having no momentum implies that its energy is also zero and this simply means that there is nothing. 
A: As per our currently accepted model, the SM, there are elementary point particles, with no internal structure, no spatial extension.
There are two types:


*

*massless, like the photon, gluon

*particles with rest mass, like the electron and quarks
Now the rest mass of the electron is an intrinsic property, theoretical, calculated, nobody actually measured an electron at rest.
If you put a photon into a box with massless walls, the box gains rest mass. Why? because the photons in it are exerting pressure on the walls. 
Now matter, is built up by elementary particles. Quarks, and massless gluons build up neutrons and protons. Only 1% of the mass of the neutrons and protons comes from the rest mass of the quarks. 99% is the gluons (massless) bonding energy, because they are confined into the neutrons and protons (like photons in a box).
Neutrons and protons build up nucleuses. Some of the mass of the nucleus comes from the bonding energy between the neutrons and protons (massless pions bonding energy, like photons in a box). The rest comes from the rest mass of the neutrons and protons. As per the correct comments, as you go to higher levels (nucleus, atoms, molecules), the binding energy actually becomes negative in the mass of the total QM system. 
But the main point is, that these nuclei's rest mass is mostly from the binding energy of the gluons inside the nucleons. 
Atoms are built up by nuclei and electrons. The rest mass of these adds some part of the atom, but the bonding energy is again there (actually it is negative).
But here too, most of the rest mass of the atoms are made up of the binding energy of the gluons inside the nucleons.
As you build atoms into molecules, the rest mass of the atoms is there, but the covalent bond adds to the rest mass of the molucule too (negative). 
Here too, most of the rest mass of the molecule is made up of the binding energy of the gluons inside the nucleons.
As you go closer to the macro world, less comes from bonding energy (confined massless particles like the photon in the box) and more from the rest mass of the constituents. The closer to QM, the more comes from the confinement of massless particles. 
In relativistic QM, gluons are traveling at speed c. Their bonding energies give a lot to the mass of neutrons and protons. You are right, SR is compatible with QM.
Mass at the lowest scale is mostly just these massless particles (moving at relativistic speeds) confinement (bonding). Quarks rest mass adds only very little. As per our current model, the SM, these quarks and electrons do have an intrinsic rest mass (that we explain by Higgs). Maybe one day when we will have figured out what these are made up of, we will know that the rest mass of these are just confined massless particles too (strings).
