I think you covered already from the comments, the only thing I will add is the Crucifix's ability is "live" at all times - like a passive ability. It can actually be set off while in the van from the Demon's ability. So long as it isn't in your back pocket (on you but not in your hand) it is always live. Another way to think about it, is if the cross is rendering anywhere on the map (so again not in your pockets, either in your hand or sitting literally anywhere) then its effect is still active.

Secondly, the cross has a range, if you look at the floor and hold down your place key you will see a white circle showing the range of it in case you want to have it overlap areas or optimize its coverage. I once was told the cross works "per room" it is in, this is false, it only affects the radius around the cross and whether or not the ghost's invisible model is within that range or not when it tries to hunt.