Half-halt in VERY simple terms: drive, squeeze (hand), release.
I was told by well-known trainers and Olympic level riders that the release is the MOST important part. It's the release phase that builds self-carriage.
These three bits are tiny adjustments, a little close of the leg or slight driving seat; a tiny closing of the hand (tightening the spring or the bow, if you prefer), then the release of the "spring".
The more highly trained a horse is, the smaller and lighter these things should become. The seat can become either ******ing or driving (whatever you need at the moment), the hand softens as the horse rebalances in collection, sitting more on the haunches and lifting the forehand.
When it's working right, it's a pretty cool sensation!
Remember that the half halt given on the right side influences the left side ( can be felt most at the trot). Also as I said you cannot influence a grounded leg.
So the term cycle of energy going from the hind end through the back to the poll and to the hand is in fact the half halt affecting the swing to the hind leg so that the forward thrust energy is converted to upward support energy.
This lifting of the leg does not influence the timing as it takes just a long for that leg to travel upwards and take the down step as it would for a normal forward stride. But the stride will be shortened ON THAT ONE SIDE.
Therefore just throwing half halts at a horse at random will not do what the rider may want and when a coach constantly shouts half halt half halt half halt at the rider, that coach either doesn't know just what they are trying to achieve or they hope one of those half halts is done at the right time ( sort of hit or miss).
The problem is that so many coaches are taught what a half halt DOES but not how. The timing is critical to get the best results ( again because of when the hind leg is in position).