This word is significant in contrast with the preceding relation words
because it can only mean one thing. As we five sat at table over lunch
that Peggy at one point turned her finger to her chest, pointing, and
said “me”. I don’t recall the details of context but remarked that her
use of “me” was unequivocal and standard.

One would not expect Peggy to recognize that “me” in another person’s
speech is reflexive… or maybe she would. (Try “me have that”
tomorrow.)