One of the problems is that the only published procedure for a standard overhead join depicts the aircraft approaching from the live side of the airfield. Exactly what the sequence of events should be if approaching from the dead side is undefined.
Yes, inexplicable that none of the diagrams show this, leading to significant confusion.
For example, should such a joiner cross the centreline to the live side while still 1000 ft above circuit height, before returning to the deadside in a left hand turn?
That's what I was taught.
Or is it acceptable to descend from to circuit height remaining on the deadside? In which case what's the difference from a crosswind join?
I think that's a crosswind join, as you suggest, not an overhead join.
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.
So if you approach from the dead side, you do a 180 to return to dead side, another 180 while descending to join crosswind, then a 90 to turn to downwind, a 90 to base and a 90 to final. A dizzying 630 degrees in all.