Radiative Flux, Irradiance and Luminosity confusion
It looks to me like irradiant flux is the same as radiant flux is the same as irradiance, both are energy per second divided by area.  That interpretation would seem to be consistent with everything you quoted.  In more precise terms, these should all be "flux density", not "flux", because flux just means the amount of something per time, but that convention is never followed much any more.