So there's 2 aspects to this

What cops theoretically could do if they're properly trained and motivated and working on the crime of the century with the media and mayor's office breathing down their neck

And there's what they're actually going to do for anything else.

Theoretically it's almost impossible to be truly anonymous in the world we live in today. If you make a phone call, there's phone records the cops can get access to, security cameras everywhere, if you call from a deactivated cell phone or take out your SIM card they can try to get the the IMEI number and see who that phone was last registered to, if you submit something online they can try to trace your IP address, etc. they can try to track down witnesses who may be able to ID you, etc.

Basically none of that is ever going to happen just to trace down a witness that called 911 who's probably not going to be cooperative anyway. People watch too many CSI TV shows.

And good luck trying to get cops to try getting fingerprints for anything short of murder, and even then they're going to be looking for the suspect, not a random passerby who called it in. They're also probably not going to get useful prints off a payphone because 10,000 people have probably had their fingers all over that phone since the last time someone bothered to wipe it down. And if your prints aren't already in the system, they're not going to be able to tie them to you anyway (although from the way you're talking about it I suspect a lot of people probably have previous records and been fingerprinted)

The practical answer is call from the payphone, call from a deactivated phone without a sim card, call from a borrowed phone, call from a TextNow or similar service number. That is more than enough anonymity that in all but the most extreme serious crimes the cops aren't going to put in any real effort to try to track you down as a witness. They may put in a little more effort if they think you're a suspect, but it's usually pretty clear if that's the case. Most of the time, they're probably not even going to bother looking up the phone records even if you call into the station from your own number.

If the department has it, you can also try an anonymous tip line or submit the tip online. Those may not be checked very often, so I wouldn't necessarily count on that if you have an emergency you want to be addressed quickly.

Really, just call, they're not trying to bust people for other stuff for calling in an emergency, they're not ratting people out as the one who called, that's all counter productive and just makes more work for themselves in the long run. Make the call and leave the area if you want, we can't make you stay there and you should have at least a few minutes to skedaddle before the cops show.

Edit: call from a nearby business' phone or a borrowed cell phone, and just don't leave your name and try to stay off a security camera while you're doing it. Unless you're really distinctive looking, odds are the person who let you use the phone is probably not going be able to give much of a description of you if cops ask for some reason