Basically, there are two schools of thought. An outdated one, which merges γ into m, and the current one (e.g., the Landau lineage) which leaves m alone. You won't find any "relativistic mass" in any decent source published since the famous theoretical minimum ( https://en.wikipedia.org/wiki/Course_of_Theoretical_Physics ).
There is a very good reason for this approach. In the simplest, most classic way of deriving the special relativity from the first principles, the 4-velocity is introduced before the 4-momentum. I.e., γ and m are coming from different places and are not connected in any ways whatsoever.
"In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest". This is the case of a system being heated. I'm still led to believe that the mass is still increased by the increased motion of the particles in that system. https://en.wikipedia.org/wiki/Mass_in_special_relativity
As I said, algebraically both approaches are equivalent, so there is not much harm in using this notion. But the "relativistic mass" does not have physical meaning and does not make any sense because of the way relativity theory is defined. And I would not count wikipedia as an authoritative source, anyway. 2nd volume of the Course is a bit more legit.
There is a very good reason for this approach. In the simplest, most classic way of deriving the special relativity from the first principles, the 4-velocity is introduced before the 4-momentum. I.e., γ and m are coming from different places and are not connected in any ways whatsoever.