I think I can disambiguate a little bit, as I had a flat tire on freeway just this morning.
When the tire pressure is low, the system advises just that: low tire pressure. According to my experiments that happens when the pressure falls below ~32 PSI.
Now, the “Service TPMS” alert comes up when there is actually something wrong with the system, such as when the system can't get the signal from one of the sensors. When I replaced the flat with the temporary spare, and placed the disabled tire in the trunk, where it was not spinning, my low tire pressure warning got replaced with the service TPMS warning after a few miles of driving on the spare.
Once the flat tire was patched up and put back on, the system got green (or should I say gray) just after a couple of minutes of driving.