In this case, I found the KAP 140 autopilot found in my 172S trainer non-intuitive, and a cursory reading of the Supplement wasn't enough; so I decided to dig in and extract the essence into a more easily-understood diagram.
It's important to realize that there are (to my knowledge) three different hardware variations of the KAP 140:
The diagram here refers to the last of the three hardware variants with the service bulletin applied. Also, I've only incorporated modes relevant to VFR (so the APR and REV lateral modes are not shown).
The green rectangles indicate independent ‘state machines’. Specifically here, when the autopilot is engaged the lateral and vertical states are both active and operate independently of one another; so laterally you might be in ROL mode and vertically in the ALT mode. This also hopefully clarifies another point of confusion that I had, which is that it isn't possble for only (say) the lateral mode to be active and not the vertical mode.
The solid line arrows represent state transitions: usually these are triggered by button pushes, indicated either by the tail end of the line being connected to the source state, or it floating within the enclosing state machine in the case that the transition can be triggered by any state in that machine. The lines with a ‘flat’ tail that is disconnected from the source state indicates transitions that happen for reasons other than an explicit user action (button push); for example, the ARM states are exited automatically by the system when given external conditions are met.
The dotted lines are intended to show triggers that are not state transitions, although I agree that this concept isn't yet clearly developed. For example, when the Vertical Mode ARM state is exited, there is a dotted line that attempts to suggest that the Alerter Reference Altitude is copied to the Hold Reference Altitude.
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