So I interpreted this question differently to most other people here, interestingly enough.

So when you call someone, on your headset, you hear a ringing dial tone thingie as the phone "rings" on their end. The question is: If their phone is on silent, do you still hear that noise on your headset?

I imagine so - it's probably used for many automated systems to detect when you pick up or similar.

You mean what’s the vernacular?

“Ring” is a hold over word from when phones had bells that rang upon receiving a signal from an operator. Now it means anytime a phone makes a sound to indicate an incoming call.

If a sound isn’t made, you can just say “notify”, because chances are it creates a dialog, or flashes a light, or does something to notify you that someone is calling your phone.

Depends how literal you want to be.

In the loose sense of, "Someone is trying to make a phone call to me and my device has registered that fact and is now alerting me in some way, whether through sound, light or movement" yes I'd still just say it's "ringing".

It vibrates

the screen turns on

Yeah. Put your phone in a vacuum.

I refer to a phone notifying you of a call - with a ring or otherwise - as "going off." However, after looking this up, that apparently means "to explode" (possibly metaphorically, depending on which source you trust).

So far, even on vibrate or silent, my phone has not done that.

After processing the question while writing my replies in this thread, I'd say it doesn't ring. If it vibrates then it vibrates, simple as that. If it neither rings nor vibrates but the screen turns on, then... it flashes? Not super sure about that one.