iPhone text notifications not occurring when near iMac or Watch

Hi all, just recently switched to my first iPhone from Android, coincidentally also won a gen 2 Apple watch in a contest so it is connected as well.

I haven't figured out the specific combination of settings and/or devices being in proximity which cause this, but my phone seems to not notify me of incoming text messages when I'm at my computer. I receive texts for work so this is a problem of course.

At first I thought it was the watch trying to take over notification duties, and it may still do that either way regardless of me setting it to not mirror iPhone, however I just turned it off, had someone send me a text (non-Apple user), no notification. Open the phone up, there it is waiting. I've noticed that when the messages are to/from Apple/iCloud users, they show up in the Messages app on my computer, and if my computer is not muted, then I also get a notification sound from the computer.

This is making me think that when my phone is in the proximity of my computer, the iCloud stuff is letting them interact in a way that the phone is expecting the computer to take responsibility for notifications for incoming texts. That's of course a problem for two reasons, one, if it's on mute, no notification occurs, and two, if it's from someone not on iCloud, Messages.app never sees it even if the computer weren't muted, so it just sits there unbeknownst to me.

Can someone validate if it actually is the computer causing the issue? If that does prove to be the case, is there a combination of settings I can use to have my phone always notify in the manner it's been told to regardless of what other devices are near by?

Thanks!