Thanks for that description. The “stop” behavior you describe is different than what I’ve experienced all this time. Tapping “Stop” (or swiping down from the top of the watch face, which is much, much easier) stops the alarm for the day, and tapping “Snooze” keeps it active but sounds again at 9 minutes.
I think the difference is you may have set up a schedule in the Sleep app, so your alarms recur on that schedule. I do not have such a thing set up (I like to sleep, but there’s only so much about my life that I want to hand over to Apple…or maybe there’s so little that I haven’t!). So when I “Stop” an alarm, the Watch remembers that alarm but it is now inactive unless and until I turn it on again. Since my schedule is just irregular enough that I’m never really sure which of 3 or 4 alarms I want to set for the next day, I activate the one I want at night.