Question about effective energy in stochastic thermostats

My questions are related to the point you mention about the existence of a conserved quantity for your thermostat and Langevin thermostat. My first question is why you call it effective energy? Is not an energy in a well-defined sense? is it OK to interpret it as the energy of the system plus an “energy of the thermostat”?
Another point is that usually the codes that I used (NAMD with Lanvegin, Gromacs with both,…) provide as output a lot of data such as energies (dissected in different contributions), pressure, temperature, and so on but not the “effective energy” which could be useful to track the correctness of the integration scheme and selected time step. Is there any reason for that?
Thank you very much and thank you for your great talk.

@bussi could you answer? Thank you