NOT A TEACHER
(1) Great question!
(2) These are only my opinions. I do not claim that they are correct.
(a) Some books say:
(i) I remember him being there. = I remember that particular person. (I remember
HIM, but not HER, being there.)
(ii) I remember his being there. = I remember his presence there. (Yes, police officer,
he WAS there.)
(3) IMHO, "I remember that he was there" is ambiguous. It could be either 2ai or 2aii.
(a) Maybe (repeat: maybe) it depends on how you say it:
(i) I remember that HE (not SHE) was there.
(ii) I remember that he WAS there.
Student or Learner