I have never experienced that behavior with my watch (Series 11, OS 26.6) but when I use Siri, I say either "Siri" or "Hey Siri," then I pause for maybe about a second, and the rainbow microphone appears, then I tell it what I want. It always responds to that command or request when I do it that way, without the greeting. With that very short delay after "Siri" or "Hey Siri" I don't get that greeting. I do get the greeting if I say "Hey Siri" and then say nothing after that. So try "Hey Siri, [0.5 second pause], put milk on my grocery list"
All of these goofy pseudo-robots that are voice activated work a little differently. Alexa has all kinds of quirks as well, just different ones.