Watch notifications while watch is ON wrist... while iphone is locked

Iphone 14 pro. IOS 16.0.2, apple watch Ultra with cellular plan.

• Am I reading this correctly that the only way I can get txt notifications on my watch is if your iPhone is locked or asleep AND your Apple Watch is unlocked (regardless of whether the watch display is asleep or awake) and on your wrist, you will receive notification alerts on your watch.

Is there no way to get the benefits of wrist detection AND as long as your phone is locked, send notificaitons to your watch/iphone at the same time? Does this mean if I leave my phone at home I won't get any notifications because my phone is locked at home as is my watch while i'm on the move?

In the even I don't have cellular, It seems reasonable to me that your watch should have the capability to recieve notifications while your phone is locked. That's why I have my watch... to notify me, so I don't have to use my phone.

I understand I can do this if I disable wrist detection but then I get double notifications all the time.