The game registers cuts based on the number of wheels out you set on the server. As far as I'm aware it should work with four wheels set, although admittedly we nearly always race with three wheels out set. Unfortunately we only have the information the game gives us, so if the JSON file doesn't have the cuts in it, then we don't have any information about cuts.
And yes, when a lap with a cut is faster than a driver's average the following should happen:
View attachment 295285
As you can see, this drivers second sector and lap are not set as their fastest lap because they cut the track, even though they were faster.
The only other variable here is the track itself, some tracks (usually mod tracks, but some official tracks have this problem too) have very wonky ideas about which surfaces are legal and don't always detect a cut when they probably should.
I'll try to do some testing with four wheels out set when I have some free time, although if it doesn't work properly then there's not much I could do to fix it, as it sounds like the game itself is the problem here.
Thanks