The average energy is a weighted ledger: each state's energy is multiplied by its probability and the results are summed. It is the standard expectation value, the center of mass of the energy over the probability distribution. The arithmetic is exact for the stated weights and energies; real measurements need energy calibration, temperature and model choices, and carry statistical and systematic uncertainty.
The average energy is a source-bound weighted sum. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.