If you are not being radar vectored to the FAF or on a “no PT” transition, are there any other circumstances where a procedure turn that is depicted on an approach chart is not required?
First, I don't believe you need to be vectored to the FAF, just the final approach course, in order to forego a published procedure turn.
Also, you could be cleared "straight-in" when ATC gives you the clearance for the approach. It may seem subtle, but it makes a difference. Here is the definition in the pilot-controller glossary:
"STRAIGHT-IN APPROACH IFR. An instrument approach wherein final approach is begun without first having executed a procedure turn, not necessarily completed with a straight-in landing or made to straight-in landing minimums."
Also, don't take it literally. If you approach at a 45 degree angle to the final approach course and are cleared "straight-in", you still approach at a 45 degree angle; there's just no procedure turn.
Using the below approach as an example here the difference would be, "Cleared 0M3 RNAV RWY 2" (Must do the PT) vs. "Cleared straight-in 0M3 RNAV RWY 2" (Shall not do the PT).
Another indicator I know of is notation on the MSA ring that declares a range of radials of approach as "No PT" such as this one. Note that the MSA and radius on both halves is the same, so the only reason it is split is to declare no PT on the southwest approaching side.