Rehearsal..who directs it?

Is your officiant attending the rehearsal? If so, I would think that they would lead it. Otherwise, is your day-of coordinator attending the rehearsal?

Our officiant directed the rehearsal.

If it's in your DOC's contract, they're normally the ones who will walk you through the rehearsal (where to line up, in what order, how to walk in, etc). Your officiant will be there as well to walk you through what you need to do for the ceremony. Then, your DOC will direct how to leave the altar (again, what order, etc.). Look over your contract and if it's not in there, ask your DOC if that's in her scope of work. Then, double check that your officiant is able to be there as well.

The officiant normally does or you can ask the day of coordinator

We didn't have a wedding planner or day of coordinator, but our venue had a person that kid of acted like a day of coordinator so between she helped direct and did our officiant.

Our coordinator was supposed to do it but she wasn’t there last minute, and her replacement was entirely incompetent and giving people the WRONG directions despite the instructions for how the ceremony would go written on a paper right in front of him lol... so I ended up running it myself

Our coordinator ran it, though I expect the officiant would have, if we hadn't had a coordinator.

Our planner/DOC coordinated our rehearsal. I would ask your DOC if that is included in her contract. If not, you might be able to add that service on. Otherwise, you officiant might be able to do it.

Our officiant ran it and the venue coordinator assisted. Either the officiant or DOC is usually who runs the rehearsal, though you should check with them to make sure.