You're talking about relativistic mass, but that isn't a commonly used concept anymore. Today it is indeed the norm to think of "mass" as being the mass at rest. Massless particles are really massless and have momentum without having mass.
Yes, using it as the relativistic mass leads to confusing papers and equations, so it's better avoided. Yet, the press does that, what makes everything very confusing for lay people. I should have clarified that the OP usage of the term wasn't optimal (although common on the media he sees), but this following discussion about word definitions seems to be completely missing the point.