When I did that once at Cambridge, ATC had a serious grumble. They made it clear: OHC means pass overhead (or with overhead on the circuit side of you - e.g. if LH circuits then pass with the centre of the runway a bit to the left of you), then turn (if necessary) in circuit direction until you're facing the dead side, cross to the dead side, and only then descend to circuit height while turning in circuit direction.
Interestingly, I was in the circuit at Cambridge 10 days ago when ATC had a grumble at a joiner who joined crosswind from the deadside when asked to join overhead. It's clear that Cambridge ATC (for whom I have a great deal of respect) thinks there's a difference, but I have a lot of sympathy with those who aren't on the same page.
Personally, I think there's very limited value in an overhead join at a controlled airfield.