We always thought Gotham was Manhattan, but it turns out it's actually Seattle.

Gotham is so perpetually fucked by so many supernatural phenomena that it's actually rare for that to happen

Wouldn't it work the other way, too? If the bat signal is off, it's crime time, my fellow criminal goons!

In the Adam West TV show Gordon just picked up the red phone.

Batman checks his phone: "Fuck."

Since Batman doesn't wear glasses and has a mask, I feel like it's just as likely he'd have a like a smart glasses type thing built into his mask. So he really wouldn't have to look anywhere but the notification would just be a translucent overlay.

Or maybe full coverage heat retardant smart contacts, so they protect his eyes from any debris/explosions. Kind of like what Tom Cruise wore to perform the outside the plane stunt in the last mission impossible movie. Covers the whole outside of the eyeball so the wind does dry out your eye or make it so you can't see.

Didn't see the movie, but heard about the stunt

In the movie The Dark Knight, Gordon says Batman often doesn't show up when the bat signal is on, he mostly turns it on to to remind criminals that Batman is out there.

In the comic The Dark Knight Returns, Gordon says there's about a dozen different ways Gordon could contact Batman, but he uses the Bat Signal for similar reasons.

So it's mostly just for theatrics, if it were a real emergency Gordon would phone up Batman. But I guess it's a way to indicate to Batman "let's talk whenever you're not busy".

[...] and mostly ignores it

That's not where my gutter brain went after reading the first five words. Where my brain did go was a Hans Niemann scenario.