How does the Greenspan/Bernanke Put semantically relate to put options?
then rational investors would buy call options or sell put options, not buy puts. So how can a Greenspan or Bernanke Put be sensibly analogised to a Put Option?

Investors do not need to buy or sell anything, the story is that the Greenspan Fed was giving out free puts to investors.
It doesn’t literally work that way, but it is directionally close  to making sense.