Jose wrote

but do you have an
official definition of "rating limitation" to use to properly
understand 61.31(k)(2)(ii)?

No, but the definition of rating is implicit in its use in that same
section. 61.31.... (a) type rating required... (c) aircraft category,
class, and type ratings... (d) aircraft category, class, and type
ratings... and then (e), (f), and (g) specify "training", not
"rating".

Anything pertaining to "rating" is a certificate action accomplished by
either an FAA Inspector or a Designated Examiner.
Anything pertaining to "training" may be accomplished and endorsed by
an Authorized Instructor.

Bob Moore
Authorized Instructor