I chose the two scenarios I did to illustrate that "frames per second" is clearly not meant to be measured in terms of times the display refreshed, but rather in terms of times content was actually rendered by the game engine.
In my opinion it is quite difficult to provide a definition of "fps" that somehow makes 45-fps-native-with-frame-doubling be counted as 90 but doesn't also make either of the ludicrous examples I presented be counted as 90.
I understand now, but I think any full frame that comes out of the GPU frame buffer is a frame. A real rendered frame or a generated frame using some algorithm. Even in the silly "I duplicate each frame" example, you are outputting that number of FPS. If you stand still in a game and nothing changes in the frame you're still counting all those practically identical frames.
A measure for "FPS effectiveness" sounds interesting. Like how much detail, changes, information can you discretely convey per second relative to what the game is continuously generating.
A Nyquist of sorts. Are you just duplicating samples? Are you sampling a high frequency signal (fast motion in the game) at high enough rate (lots of discrete FPS)?