Are joist brackets supposed to hang under the board or sit on top?

You've got it backwards, not them. Joists sitting on top of the beam is the standard way to frame a deck, and it's actually the stronger setup. The hardware you're seeing isn't doing the job you're picturing. It's not carrying the weight at all, the wood-on-wood bearing is doing that. Those brackets are just tie-downs, nailed to the side of the joist to keep it from lifting or shifting off the beam in high wind. Uplift is the word for that force, and side-mounted clips are exactly how it's handled when joists bear on top of a beam. The setup you had in your head, hardware cradling the wood from underneath, is a joist hanger. Those are used when the joists hang off the side of a ledger or a flush beam instead of sitting on top of one. Two different framing styles, two different pieces of hardware. Since your joists rest on the beam, hangers wouldn't even make sense there.

One thing worth checking while you've got the surface torn off: make sure the nails in those brackets are the fat, short ones meant for that hardware. It's really common for someone to have used little finishing nails or drywall screws in ties like that, and those snap or pull out long before the bracket itself fails. If you see shiny short nails in there, swap them for the proper structural nails when you reassemble.

Also a good window to run your hand along the top of the beam and the tops of the joists where the boards sat, since that's where moisture sits and rot starts. If anything feels soft or punky, now is the time to deal with it, not after the new decking is down.

So no, whoever built it didn't do it backwards. That's textbook framing, and the brackets are doing exactly what they're supposed to.

Did the same thing to my own deck two summers ago, tore the boards off and had that exact moment of doubt looking at the framing. Mine had twist straps instead of the wrap style you're describing, but same idea, same nailing pattern. Mine called for 26 of those ties total and I remember counting them out twice before I figured out why. One thing I'd add since yours is opened up, check wheather the joists are crowned upward or downward. Lumber bows slightly and you want the crown up, so any joist sagging at mid span is worth flipping before the new boards go down. Cheap fix now, annoying bump later.

Ha, I had the same wake-up call on my dad's deck. Once it clicked that the brackets were just keeping things from lifting, it all made sense. Since your boards are already off, run a straightedge across the tops of the joists before you lay anything new. Ours had a couple that had sagged over the years and you'd never see it until the decking highlights every dip. Shimming or planing those flat while everything's exposed is way easier than doing it later.