You know what, you're right. And the act of putting in a number can't be called 'dialing', because there isn't a dial. Also, when a car stops at a designated spot and is shut down, that's not 'parking' because there isn't a park there like there used to be when that word was first used.
Language cannot possibly evolve in these obscene ways!
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.
“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.
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".
(of a telephone) produce a series of resonant or vibrating sounds to signal an incoming call.
All sound is vibrating, because sound is just air vibration, therefore any audio at all produces by a telephone fits this definition. I will die on this hill.
... Or perhaps the definition is just a bit outdated, only accounting for ringtones that sound like bells which used to be a popular choice. IMHO the definition should be updated to include any sound that alerts of an incoming call.
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.
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.
I also say “your phone is going off” as it can refer to either the alarm, incoming call, or other notifications. It’s definitely an accepted alternative meaning for the phrase.
"Going off" can also mean "be in the process of spoiling / going bad" when referring to foodstuffs, which I only mention to point out that the phrase has at least one alternative use beyond exploding, even if alarms and such aren't considered. More usually in the past tense of "gone off", but it gets some use in the present tense.