If it were mine, I would start by trying to understand the gist of diagrams on page 21... page 23 with emphasis on the first diagram on page 21. Note that the piston slides left and right, and lets high parts of the piston seal or not seal against seals. When a piston valley lines up with a seal, there is an intentional path for water.
My theory is that one of the seals that should be blocking water in the service mode is passing water.
Once I thought I understood the diagrams to some level, I would either get a rebuild kit and try to rebuild, or I would first carefully go inside to look for damage. On any seals/O-rings that I install or reuse, I would apply food-grade or plumber's silicone grease. I got MolyKote 111, which is thicker than people most use. A little goes a long way. I use nitrile gloves when applying it. Get silicone grease on order or get it at your plumbing supply place if you don't have it. I would rather use that grease where it is unneeded than to not use it when it might help. If anything is damaged, and you don't already have the kit, then pause while you get the parts or put it back together to let it behave as it has been, until I could take it apart once the parts come. Before going inside, I would watch the video a few more times. Remember that reusing O-rings is against the advice of the video. Getting a rebuild kit and replacing stuff, even if you don't see the flaw, is the better course of action.
Or just have a water conditioning pro who rebuilds that valve rebuild it for you. It might take an hour or so vs the 20 hours it would take me if I am lucky. He will have the parts.
I have been into my valve softener valve. I have not been into my filter valved. Neither are similar to your valve.