Update on what I think I was seeing, and it's still happening, it appears, and that is if Messages isn't open or my MacBook Air isn't 'awake', it doesn't see the message deletion that's done on my iPhone. It also doesn't catch up once it wakes up or I open the app. Right now, I have a message from Redbox and (2) messages from Amazon on my MBA in the Messages app. They were deleted on the iPhone earlier today.