I think that Ookami event threw everybody off track. Prior to that (or excluding that event as canon), we had this:
1. Constructs are storage units designed to hide Exiles between iterations of the Matrix. This is how the Merovingian & Co. escaped the Smith ruckus. (Although it never explained 01 and 10).
2. Exiles are manifestations of self-aware programs, manifested visually by archetypal / mythological forms. But before they became self-aware and started to do their own thing, they would have had these forms to begin with in case a bluepill accidently saw one of them doing whatever that program was supposed to be doing normally. So a bluepill would tell his friends "OMG I saw a ghost!" and the other bluepills would say, "Yeah whatever dude" and dismiss it so it wouldn't cause all sorts of problems.
It just seems pretty inefficient to me to have a whole iteration of the Matrix based on one sorry mix of Van Heflin / Torchwood / X-Files / Charmed / World of Warcraft etc. 