The confusion is probably that Shimano makes three cage lengths:
Short (In Zee form these come as two different connector lengths too, Fr and DH, although the cage length is the same).
“Long” (GS) – which is what most people refer to as medium.
“Super Long” (SGS).
The rear will tell you what it’s maximum size is (biggest front + biggest rear) but normally med will be fine.
Not quite right I think. The rear mech number refers to how much slack the cage can accommodate. It doesn’t really matter whether it’s on the front rings or back or whatever. E.g. the amount of slack in an 11-36 cassette is labelled as 25 teeth. If you have a 14t difference between your two front rings (E.g. a 24t and a 38t) then the mech needs to accommodate a chain growth potential of 39t.
Shimano Medium (or ‘long’ / GS as above) will happily accommodate 2×10 never mind 1×10. Don’t overcomplicate it though:
Short cage = 1x
Medium cage (GS) = 2x
Long cage (SGS) = 3x