Question: We had a HydroRight in the toilet for several years (with the split, dual flush handle) and all of a sudden, water began trickling into the bowl, and the tank would refill every 10-15 minutes, which would take just a few seconds.

I thought maybe there was a rebuild kit for the HydroRight so I could replace some seals, but didn't find one. So, instead I bought all new internals, but this time a Fluidmaster Dual Flush. This one came with the angled adapter to change the bottom mounting point of the overflow tube from an angle to being level, onto which the complex flushing mechanism fit.

The toilet still had the same problem. It seemed to me that the leak must have been where the flushing mechanism attaches to bottom of the overflow tube. The new one attached via what I believe is plumber's putty. I thought maybe I hadn't cleaned the ring adequately where contact occurs or maybe didn't dry it well enough, so I took it apart, and threw away the putty that came with the Fluidmaster. I bought new plumber's putty (which is a very different consistency from what came on the Fluidmaster adapter), rolled it out, and, after cleaning the adapter and the contact point at the bottom (where the flapper would normally touch) very thoroughly and letting them dry, applied the putty to the adapter, then pushed it firmly into place. Then, I shoved the Fluidmaster flushing mechanism back into place on top of that adapter.

Just to be clear, when I cleaned the contact point at the bottom of the overflow tube, I observed that it appeared to be smooth, undamaged, and in perfect shape. I would've expected a good seal.

Still leaks. No idea what to try next. Suggestions?

Answer: 
It looks like you might have installed the valve too deep in the wall.

The tile goes behind the silver escutcheon and up to the gray plastic circle that you can see from behind.

Usually the face of the gray circle template is at the same plane as the face of the finished tile. What kind of rough in valve is that? We can look at the instructions and it'll say. 

The round hole in your backer board isn't too big, that won't be a problem. The problem looks like you recessed the valve too far into the wall, and now the tile won't slide in between the silver escutcheon and the backer board.
That is not how it works and the trim goes over the tile for leaks and the tile installation would be easier.