How I have seen share boarding work is 2 people technically own the horse and split the cost of board/riding/other stuff. Normally done between 2 people who know each other well anyway.
Half leasing is where 1 person own the horse and someone else pays to ride the horse/use them. Sometimes it's only paying for the rides and sometimes the work is split in half as well. Sometimes if a horse is injured a half lessor may have to help pay for the vet. It really just depends on the contract.
The main difference is who owns the horse. 2 people or 1 person.
I lease my horse out while I went away and I half leased my horse to 2 people who knew each other very well. They split the board between them and I covered any costs that may arise due to injuries or whatever.
Share boarding is more like 2 people own the horse. The 2 owners split the cost of EVERYTHING in half so it's all equal pay. And then you both get to ride when you want and normally you work it out with the other owner to see who rides on what days.
And half leasing is where the leaser ONLY pays for their rides. The half leaser will normally pay around $300 a month (but does depend on what the owner wants and the price may be reduced for some people if cleaning the stall or other chores are done) then the owner will have to pay for everything else, vet bills, farrier, etc. none of that is split with the leaser. Normally the owner will use the money the leaser gives them for board or for farrier/vet bills. And as for riding normally the owner will either tell the rider what days they can ride or they work it out together that way, yes, both the leaser and rider both ride the horse.
I hope this helped!