What that guy told you is utter bullshit.
The triangulation based on signal power from different antennas, works pretty good but has a big error, kind of 100 meters in a big city with plenty of antennas near you, if there are not many antennas around the error can be pretty fucking huge, like kilometres.
Some commercial applications use that, google maps is one of them.
Civilian GPS has a precision of 20 meters and military ones 5 meters, many cells have GPS but don't broadcast their location, at least not officially
A few years ago, the notorious hacker Kevin Mitnik was being chased down by LE, and they knew the zone they guy was operating by triangulating his cell, but that gave LE like 3 blocks where the guy could be, so what they did is to go with multiple vans to drive around the zone and triangulate the signal, and that way they were able to catch the guy.
Also, as far as I know, a turned off cell doesn't transmit anything, I heard many times of LE waiting for a cell to turn on to see where it is in missing persons or kidnap cases.
A few years ago I had some meetings and we took the battery off the phones, it was like a gesture, because its the easiest place to hide a bug, today you could put your phones in a Faraday bag to be sure its not transmitting anything.