In which case, we seem back to having two things distorting spacetime. Could mass just be the momentum of massless particles going in really small circles such that you can pack them together and call them matter?
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.