I find the term "thermal energy" to be useful, despite its
imprecision. My definition is that the thermal energy is the
part of the energy that depends upon temperature (as distinguished
from any "static" energy that is independent of T). The division
of the total energy into thermal + static is of course not
unique, and different divisions might be useful over different
temperature ranges.
One context where this comes up is the equipartition theorem.
I don't want to write simply U = fNkT/2 (where f is the number
of degrees of freedom per particle), because most systems
also contain other energy, such as energies in chemical bonds
and rest energies of all the particles. So instead I write
U_thermal = fNkT/2, with the understanding that under a limited
range of conditions, the change in U is the same as the change
in U_thermal. Then we can calculate heat capacities (dU/dT)
and so on.