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This is really something you should be asking your board house.  I am not convinced the various terms for combinations of flexible and rigid circuit boards are all that standard.  There may well be variations between processes.

Just from the images you posted, it looks like what you call "flex-rigid" has the polyimide layer extending thru the whole board.  Those then become the inner layers of the rigid part, which adds its own outer layers.

Your "flex with stiffener" process looks more like rigid fiberglass glued onto a flex board for mechanical strength in selected areas.  That could be useful for mounting, for example.

"Flex-rigid" seems like it's better if the flex board has to have components mounted on it.  A flexing board with rigid components is is going to fail.  Basically, the flex parts are flexible interconnects between rigid boards with parts on them.

I would evaluate each case separately, and not be afraid to talk to the board house about tradeoffs and options.

The one time I did a mixed flexible and rigid board was for something that basically needed to be flexible, but with a connector solidly mounted at one end.  I don't remember what marketing terms the board house used to describe the process.  The solution was arrived at after a back and forth discussion with one of their application engineers.