Although it seems the consensus is that you definitely ought not to call it a delayed reception, but an anniversary celebration, I disagree that the cake or dress would be areas for a faux pas.
I think wearing your wedding gown would be fine, especially since you said it's pretty simple. Even if it were "more wedding-ish" of a dress, I think it would still be fine to wear it again on the anniversary, unless it had a train and definitely without any veil. And of course no bouquet, but if you felt like wearing a wrist corsage that could be cute.
I honestly think practically any type or color of cake would be fine, even if it has tiers (people even get birthday cakes with tiers) or is white. I would nix having a little bride-and-groom cake topper, but other than that I think you're free to choose any tiered or non-tiered cake you like. People might spontaneously decide to give toasts, but I wouldn't push it. It might be nice for you 2 to toast your guests, though.
I think the wedding re-enactment vibes would start to flow if you did things like: tossing a bouquet, having any sort of "bridal party"... things like that.
If you have pics from the wedding itself, those might be nice to display. Other relevant photo album, mementos and such might be nice to have on hand to show, too.
Suggested wording for invitation:
Joanne and Timothy Smith
request the pleasure of your company
at an evening dinner-dance
in celebration of
their first wedding anniversary
Saturday, the fifth of May
half-past six o'clock
Angelo's Banquet Hall
Kindly reply before the fifth of April
..and of course, no mention of gifts or anything. If people directly ask, you might try to reinforce that it's not really a gifting occasion, you just want to celebrate with them. For those who show up with gifts anyhow, I would suggest subtly stowing the gifts totally out of sight so others don't feel like they "should have" brought you something and feel bad.