I had AIM-to-SMS set up so I could keep my chats going even on my flip phone, and then on Windows Mobile even if I didn't have data service. These days I can't even check Facebook unless I have LTE, if the phone says 4G I know I have no service.
No what the parent says actually happens to me too. If the sign says 4g it means there is no data service. When it shows LTE or LTE+ then the data actually works.
So I'm an iPhone user as well, and what he said is correct. The phone normally displays LTE when full LTE service is available. In my experience. when you're in a normal LTE area, the 4G display usually indicates a really bad signal, as in it can't establish a full LTE connection, so it's trying to fall back to an old protocol, and thus your data performance is really bad.
In the U.S., at least on AT&T, there is a difference between 4G and LTE. When my iPhone 6S+ switches the indicator from LTE to 4G, the data speeds are many times slower.