I received a message on my iPhone from my niece. To the left of her name is a little grey half moon- identical to the half moon that appears when my phone is set to “Do Not Disturb.”

To my surprise, when I view that same message on my iMac, that half moon is absent.

Is that half moon merely a way to advise phone addressee that the sender wishes not to be disturbed?

Operator Error!

Quite by accident I discovered the solution/fix:

On the phone-
1. Launch Messages>name of contact.
2. Press the blue "circled Info button" (top-right corner).
3. About halfway down the page is a switch for "Do Not Disturb".
4. Turn that off.