>>64248920 (OP)
self-propelled gun just means its a "gun", as in a low-angle cannon, mounted on a vehicle
artillery refers to both low and high angle weapons, but usually refers to high angle weapons in casual conversation
so the M8 scott and M7 priest are both HMC, howitzer motor carriage, since they are capable of lobbed shots
but the former is a close-range weapon designed for supporting infantry while the latter is mobile artillery
the M12 and M10 are both gun motor carriages since they are capable solely of low-angle fire
but the former is a very long range artillery piece with the latter is a relatively close-range tank destroyer