How do I silence my watch during meetings but still keep it active?

Swipe-up from the bottom edge to Bring up the control center. Scroll until you seen the twin Smile/Sad masks. This is Theater Mode. Enable that and it will keep the Apple Watch quiet until you disable it.

Use Control Center on your Apple Watch - Apple Support

You can swipe up from the bottom of the screen on your watch, that brings up all of your control panels. If you look for one that looks like a bell that’s the silent one or hit the one that is shaped like a moon you can choose how long you want not to be disturbed for. I hope this is clear enough and that it helps