The only issue is one of semantics. “Mass” does not have a unique definition; that four letter word is used to refer to different things in different contexts.
A thought experiment:
Take a closed box with an electron and a positron at the middle, ready to annihilate one another.
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| (e+)(e-) |
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Everyone agrees that the entire system has mass M equal to the sum of the box’s mass and the electron’s mass and the positron’s mass.
Let them annihilate, and pretend the walls can perfectly reflect the two resulting photons:
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| <~~~~ ~~~~> |
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If you could not look inside the box, you would not be able to tell that the annihilation had occured, even by measuring its gravitation pull (which will not have changed.) Using Ring’s sense of the word “mass”, the mass has not changed.
Now pretend the box is not there, and repeat the annihilation.
While one can still refer to the entire system, one does not need to. Subsets of this system are obvious, whereas in the “sealed box” version, one only had the entire boring sealed box after the annihilation (meaning the only natural thing one could talk about was, in fact, the entire system.) In these cases, Ring’s definition of mass is not (in my opinion) the most commonplace definition. Indeed, the general public and many subfields of physics would not casually say that there is mass present. In such situations, one can use the phrase “invariant mass” to refer explicitly to the quantity defined in Ring’s posts. (The adjective “invariant” refers to the fact that this quantity of the system has the same value regardless of the observer’s reference frame.)
When talking about general relativity or cosmology, subsets of systems are not usually what one is interested in. (E.g., It is the entire star’s gravitational signature that is of interest.) Thus, folks that deal with such things will usually use “mass” to refer to invariant mass.
In particle physics, the subsets of systems are the meat and potatoes (e.g., there are two photons now, and each has zero mass.) A particle physicist will usually use the unadorned word “mass” in a subset sense, using the phrase “invariant mass” when (s)he is referring to the system-wide quantity. And when this rule of thumb fails, context usually makes it clear which “mass” (s)he was using.
In the general public, well… invariant mass, schminvariant mass! If the system has distinct subsets, they are treated independently. Thus, an isotope that gamma decays:
X —> Y + photon
goes from having mass M[sub]X[/sub] to having mass M[sub]Y[/sub] + M[sub]photon[/sub] = M[sub]Y[/sub] + 0 = M[sub]Y[/sub].
“Mass” is by no means the only overloaded word in English or in science. As long as everyone involved understands the possible uses, it just plain doesn’t matter.
Now, regarding the OP… 
Many others have given insightful explanations about the ubiquitousness of c, but I wanted to condense the ideas into a single chain, lest the big picture be lost:
- Our universe seems to have three observable dimensions that behave in one way (a way we have chosen to call “spacelike”) and a fourth observable dimension that behaves in another way (a way we have chosen to call “timelike”.)
- Relativity describes how these spacelike and timelike dimensions affect (and are affected by) objects in them. In particular, an object’s motion in a spatial dimension is constrained by its motion in the temporal dimension (or vice versa, if you prefer). This constraint involves a parameter which is often written as a velocity and labeled c.
- Thus, in relativity contexts, c shows up because this constraint is present. In particular, the c in E=mc[sup]2[/sup] is not there because of light; it is there because of relativity.
- Now, relativity requires that all massless particles (such as photons) move at a speed equal to c. Thus…
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In electromagnetic contexts, c shows up because photons travel at this speed.
(Sorry not to credit the above posters who originally said all of this.)
- If we’re in a relativistic regime, c will appear.
- If we’re not in a relativistic regime, electromagnetic phenomena are the only things going on (save Newtonian gravity), so we still get c’s everywhere!
Thus, lot’s of c’s.