They might be triggering off the fact that you personally are interacting with the system. But if that's not the case, if they are triggering off of some sound that the system makes, then here is my guess:

If these photos are indeed a picture of the innards of the same system, then, the circuit board prominently features a JRC-27F 012-S(555) relay on the board (spec sheet here). The dogs are hearing that relay click. Perhaps they even hear a high frequency whine associated with the power circuit, something that is outside of your hearing frequency range. (Edit: Perhaps it is the high frequency coil whine of that relay!)

Why the system doesn't play the chime for 2-8 seconds after the button is pressed? Don't know. One idea is that maybe the relay is associated with that timing, and the relay is being old and sticky.