Even professors that don't like using that concept warn their grad students about it because it comes up in papers still
Not in particle physics - I've never seen that notation used: 'm' always means the invariant mass. Perhaps if you go back to the 1950's but then you are really using textbooks and not papers and again they use invariant mass. If you write momentum as 'mv' I bet you would not find a single physicist in any reputable particle physics group that would think you meant it as a relativistic expression.
Other text books ranging from Coh-Tannoudjio to multiple modern physics and university level intro physics textbooks still stick with the relativistic mass system when glossing over stuff.
Rubbish. The intro texts typically introduce relativistic mass and then describe it as an incorrect and flawed concept (often citing Einstein) and then never, or rarely, mention it again. Senior undergrad particle texts never even mention the concept nor do the grad level texts or modern papers.
And the Lagrangian for that is Lorentz invariant, but that doesn't mean energy is invariant under Lorentz transformations...
Correct but a particle's mass comes from the energy it has in its rest frame which IS an invariant quantity and which is determined by the Higgs coupling for fundamental particles.