JustKate, I deeply respect your contributions to the Forum, but when you say
I have to voice a contrary opinion: You could not
do worse than "Saxon genitive".
"Saxon" is a misnomer, for reasons cited by others.
"Saxon" has very little meaning for most English-speakers;
for most of those few who have heard the term, it means an "ancient" tribe that has nothing to do with today's experience.
"Genitive" has very little meaning for most English-speakers;
for those who have studied a foreign language with case inflections it means something foreign, un-English.
So "Saxon genitive" is a doubly opaque term.
The term "apostrophe-s" is practical: it tells you what to do when writing.
The term "possessive" is meaningful to English-speakers because most of us know what "possess" means.
The term "Saxon genitive", according to the Google Books Ngram Viewer
(GBNV), in today's American English, has a frequency of 0.6 parts per billion (ppb).
It fares a little better in British English, with a frequency of about 2 ppb.
My suspicion is that the term "Saxon genitive" is not indigenous to the English-speaking world, but is a calque from one or more Romance languages
(in which the fine distinctions between Angles, Saxons, and Anglo-Saxons are often blurred). Hence...
mentioned by Einstein.
The GBNV puts the frequency of "genitivo sajón" in Spanish
around 4 ppb, and of "genitivo sassone" in Italian
at 30 ppb!