Why do I get a caution about a sign in from another city, immediately after attempting to sign in at home

See the following from Two-factor authentication for Apple ID - Apple Support

What if I don’t recognize the location shown in my sign in notification?

When you sign in on a new device, you’ll get a notification on your other trusted devices that includes a map showing the approximate location of the new device. This is an approximate location based on the IP address the device is currently using, rather than the exact location of the device. The location shown might reflect the network you're connected to, and not your physical location.

If you know you’re the person trying to sign in but you don’t recognize the location shown, you can still tap Allow and continue signing in. However, if you ever see a notification that your Apple ID is being used to sign in on a new device and you're not the one signing in, tap Don’t Allow to block the sign in attempt.

Providing accurate IP geolocation is more expensive for ISPs, and providing very accurate data is hazardous for internet users due to its usefulness to stalkers, abusive exes, and thieves.

In general, IP geolocation data is best assumed to be accurate when it’s in the same country.

Around here, IP addresses usually geolocate to within one of three (adjacent) US states, so fairly accurate.

If the two-factor request comes in contemporaneously with your own access attempt, it’s likely correct and should be approved.

If not, or particularly if you get deluged with sign-in requests, you will want to deny all, and to change your Apple ID password.

It would be very unusual if a hacker was able to attempt a login at exactly the same time that you did. So if the pop-up appears at the right time, it is probably valid.

In my case, the sign in location is usually anywhere from 1 km to 4000 km away.