The Move (off the charger) appears to have two power-down modes ‘standby’ and ‘deep sleep’ the latter of which is almost like a powered-off state, although it may occasionally wake itself up for updates and IP address renewals etc. To save battery, whilst off it’s charger, the Move falls into ‘standby’ mode after 30 minutes of inactivity.
 

In general/normal ‘standby’ mode the speaker will wake when the Sonos App is opened, or it’s placed on it’s charger - that feature is ‘by design’.
 

If you don’t want that to happen, you may need to put the Move into deep/sleep (power-off) mode. It needs to be off it’s charger-loop to do that - Just long-press (5+ seconds) its top rear power button until you hear the deep-sleep/power-down tone. Another option is just to simply leave it on its charger and it then does not go into standby at all.

Thank you so much for the info.  I will have to do some experimenting.  In the week I’ve had the speaker, it has never been off the charger.  And one of the first things I learned to do was to look up how to turn it off, though whether I did by pressing the off switch for >5 seconds I can’t say.  This new behavior was a surprise today.  I am not sure, but I think I have done most or all of what you suggest. 

All I want to do is for it to stay off when I turn it off, and only turn on when I turn it on by pressing the on button.

If it is always on, or if it turns on according to its own whims (or inscrutable design) I will have to control it via the app (muting, ungrouping, whatever), but that seems kind of complicated.