The IEEE Standard Definitions Terms for Antennas (IEEE Std 145-1993)
provides no definition for 'antenna efficiency' per se. A definition
is provided for 'antenna efficiency of an aperture-type antenna' in
section 2.15 of the document. The definitions are in alphabetical
order so the definition describing the ratio of power radiated to
input power appears in section 2.308 which is titled 'radiation
efficiency' (notice the dimensional metadata).

These citations from IEEE Standard Definitions Terms for Antennas
clearly suggest claims of a universal formula and definition for
antenna efficiency are incorrect. Not everyone means the same although
they use the same name. This is why it is important to define or refer
to the definitions that are used.