For an indivisible particle, probably the simplest operational definition of "mass" in your context is $\sqrt{E^2 - (p c)^2}/c^2$, where $E$ is its energy and $p$ its momentum. Equivalently, the mass of an indivisible particle at rest is $E/c^2$ in that frame.
A lump of gold and a proton are not indivisible particles, because their constituent parts can move with respect to each other. (In the case of the lump of gold, this movement comes from the thermal kinetic energy of the individual atoms in the center-of-mass frame. In the case of an atomic nucleus, which is highly quantum-mechanical, talking about the "motion of the quarks" is really an inappropriately classical picture, but it will do for this discussion.) We therefore need to be a little more careful about how we define their mass. There are two reasonable options:
(1) We could define the mass to be the sum of the individual masses of each indivisible constituent piece. Under this definition, the mass of the lump of gold does not depend on its temperature, and the mass of the proton is simply the sum of the rest masses of its three constituent quarks (since gluons are massless) - the "proper mass" you mention above.
(2) Alternatively, we could "lump everything together" (or "coarse-grain") the composite object and give it an effective mass defined to be $E/c^2$, where $E$ is the total energy of the composite object, including the kinetic energy of its constituents. Under this definition, the effective mass of the gold does increase when it's heated, because its constituent atoms gain kinetic energy. And the effective mass of the proton becomes much heavier than the sum of its quark rest masses, because it turns out that the quarks are so relativistic that their kinetic energy dwarfs their rest mass energy.
In practice, the second definition is pretty much always a more useful definition of the "mass" of a composite object. This is because in general relativity, the kinetic energy contribution gravitates, just as much as the rest energy contribution does. Similarly, it's more difficult to push something with a high effective mass than something with a low one, even if their masses are the same under definition (1). So the second definition is more experimentally accessible. Moreover, the first definition raises the question, "where do you stop?" We treated the gold atom as being made up of atoms, but of course those atoms are themselves made up of protons, etc. If we only counted the rest mass of those sub-constituents, the mass (under definition #1) would decrease further. So it's better to treat the kinetic energy of the constituent particles as contributing to the rest mass of the composite object, as counterintuitive as that idea may be.
As for a single particle moving at relativistic speeds, the unambiguous fact is that it gravitates more strongly than an equivalent particle at rest. You can tell two different stories to explain this: (1) only mass gravitates, but the relativistic motion of the particle actually increases its mass, causing the additional gravitation, or (2) mass and kinetic energy both gravitate, so the particles mass has not increased but its new kinetic energy causes the additional gravitation. Both stories result in equivalent equations, so which one you like to tell is largely a matter of taste. Special relativity was originally thought of along the lines of the first story, but now most people find it easier to keep things straight if they tell the second story. As you become more comfortable with special relativity, you'll eventually stop worrying too much about what counts as "mass" and what counts as "energy," and realize that at the end of the day they're really the same thing.
