[whatshisface]

"Mass" is imprecise enough as a word to refer to either inertia, E/c^2, or rest mass. Rest mass is indeed invariant but the rest aren't.

[semajian]

I've never come across a physicist who used the term "inertia" to refer to any of the common relativistic quantities.

[whatshisface]

I used the word "inertia" to refer to the various inertia-related things that are used in mechanics. For example, the mass tensor that appears in curvilinear coordinate systems. This class of inertia-related things are usually given a name involving the word mass, and they are not usually Lorentz invariant.