I agree that to fully specify electromagnetism you also need to include how the fields affect charged matter. So EM = Maxwell's equations + Lorentz force equation (not sure why you say there is no consensus about what this is, that is new to me).

This is just a matter of taste, but OTOH I would not include descriptions of how some materials respond to the fields in the continuous limit as part of a definition of EM.

It is true that for most terrestrial applications you do need those to do anything useful with EM. But if you want to study plasmas you need to add Navier-Stokes to EM, doesn't mean hydrodynamics is part of EM. To study charged black holes you need EM + GR, but it still makes sense to treat them as mostly separate theories.