If im not mistaken PS1 and PS2 do not output CSYNC unless modified internally so your assertion is correct. The cable must be taking Composite Video as Sync and using a sync stripper to output CSYNC/pure sync like you said.
With this in mind, you need to be mindful that most CSYNC and sync strippers output is TTL and if you're connecting to a 75 ohm input it needs to be properly attenuated via a resistor.
Connecting TTL level voltage sync to 75 ohm can have various problems such as not syncing or intermittent dropouts. Reading up on the retro console cables website he states the cable should be used on Extron Crosspoints and Pro monitors which points to it most likely being TTL level sync.
Framemeister sync input expects 75 ohms on the sync line. It is possible for your sync to not be 75 ohms and still work on it, but it varies on a console by console basis and even motherboard revisions. I would recommend you open up your scart cable and confirm the output of the sync line pin 20 is 75 ohms. Listed below is a good video on how to test it provided you have a multimeter.
https://www.youtube.com/watch?v=Z3sE3uk5kPU
Still kinda weird it works on your Fat PS2 but it is a different revision.