Part of our confusion was that they used half halt and self carriage more or less synonymously. I see a connection but don't understand them to be the same.
The half halt is a tool that maintains or corrects the carriage of the horse along with the other "tools" the rider can use ( legs, seat,voice).
Correct carriage is the result of half halts along with the driving seat and legs that guides the horse into adapting to the bit according to its level of training and physical ability.
So yes they are intertwined but not exactly the same.