We don't know those we saved perceived their surroundings.
But we do, or at least quite a lot of it, because we perceived them the same way--except for the Shades, and their not noticing that things are shifting around. Aside from those matters, what we're seeing "in town" is what they're seeing. Which is why I don't think I agree with this:
It would make sense if everyone remembers the version of the Interim that they themselves experienced
...at least, not if you're suggesting they remember it clearly. That means Kaitlin remembers the accident, losing her father, MC banging her and her mother at the same time... Kara remembers, not only the Mustang, but also the threesome with Marisa (Anita gets this in her ending, but Kara doesn't). Sammy and Bud remember being all but forced together.
OTOH, maybe they don't have clear recollection of everything that happened, but rather some vague, almost-subconscious memory of it. Maybe dreams about it (or parts of it), or something along those lines? While I wouldn't like to think that new-life Kaitlin has nightmares of almost dying in a MVA, that would explain her going into medicine. Maybe Kara doesn't see the BDSM dungeon, but sees her trips with MC in her dreams?