I've posted this before but, while there may be other things at play, this bug is also very game specific.
I was playing a lot of MLB and Watch_Dogs in May/June, and MLB would ALWAYS record the proper clip when I hit share. Watch_Dogs would work 50% of the time, the other half the time it wouldn't even have the option to save a clip. Just "Hit triangle to save screenshot" No option for hitting square for video, and I even tried pressing square anyway and nothing. Exiting the share menu and bringing it back up fixed this but of course it only started recording from the last time I accessed share so the clip was like 2 seconds long. MLB did NOT have this issue for me at all and I was switching between both games frequently.
So as someone mentioned before there may be something on the system level where it works right after an update and eventually breaks again, but the game being played is definitely a factor.