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An energy signal is defined by finite energy over all time: E = ∫_{-∞}^{∞} |x(t)|^2 dt < ∞. Because the signal exists for all time, its average power is found by looking at energy within a growing time window and dividing by the window length; with a finite total energy, this average power tends to zero as the window length goes to infinity. So the defining properties are finite energy and zero average power. Descriptions that involve infinite energy or nonzero average power don’t describe an energy signal.