But the air does move! Why you are trying to classify movement in this extremely pedantic and specific way is what is mystifying to us all.

The air particles are excited, you said it yourself. They suddenly have more energy from the speaker cone's in-and-out movement. How does this energy manifest in the air particles? They move. And when they move back and forth very quickly, in time with the frequency that the speaker is vibrating at, this causes a sound wave to travel as a longitudinal wave. How do you describe what these particles are doing, if they're not moving? They're not perfectly stationary in the air, are they? In fact, they never are, they're moving all the time. They just move in a particular way when influenced by speakers.

Nobody thinks that speakers blow air like a fan. That doesn't obviously doesn't happen, but that doesn't preclude the air from any kind of movement. Hence, speakers move air.