Mysterious toilet leak
I had a similar problem once but the leak was visible and the outside of the toilet bowl was wet. Flushing from the tank produced leakage but flushing with a bucket did not.

It sounds like clean flush water is somehow escaping on its way from the flush valve to the toilet bowl. This could be from a misaligned tank fitting or a crack in the ceramic material. I assume that you have found no water running down the outside of the fixture. 

Depending on the design of the toilet, the connection between the tank and bowl parts may include a metal fitting set into the bowl. This can become loose or cracked and thus leak water outside the flush pipe but still inside the toilet. 

Or an imperfection or crack in the ceramic could allow clean water to escape from the rim passage or wash passage, well, whatever it's called and the design of these passages varies a lot. The point is, flush water can escape its proper path and still not get outside the toilet where you would see it. 

EDIT: couple of pictures to show what I mean:




The red callouts show where an invisible leak might occur only when tank flushing and not when bucket flushing.