Oh, it’s probably worth noting also that
today(’s) acts as a deictic, which is why “our today’s meeting” doesn’t work. Deictics always add definiteness to a noun phrase, and so do possessive pronouns and determiners; and you can’t mark a noun phrase for definiteness twice (or mark for both definiteness and indefiniteness). That’s why neither “the/an our meeting”, “the/a today’s meeting”, nor “our today’s meeting” works:
today’s makes it definite, so you can’t add another (in)definitiser. (This doesn’t go when
today is used as a simple, non-deictic noun, as in @GEdgar’s example.) —
Janus Bahs Jacquet 5 mins ago