I used to use Shire horses for farmwork. The key point about ploughing is that as Marty was getting at, the horse walking in the furrow is the more important one, as this determines the mouldboard of the plough is placed to correctly turn only unbroken land. Perhaps this horse would therefore naturally get a unique name?

BTW, my favourite Shire gelding came from Yorkshire, so I had to learn Y. dialect terms to talk to him - "Gee back", "Arve" and others! The local Welsh terms were totally different and he would put on his most cunning "Wot, me?" look if I didn't speak Yorkshire.

PS Singles, pairs, and tricorn hitches are all fairly common, though a tricorn hitch would more normally be used for really heavy work like pulling a binder (precursor of the 'combined harvester' or combine).
The harness and equipment didn't vary enormously from one end of the UK to the other, but the words and terminology certainly did! For example, the spreader bar used to hitch a number of horses to one point of pull was variously known as 'swingle tree', 'swivel tree', and 'whipple tree' in almost adjacent counties. Just goes to show what we know on this board: even if you have a shovel in common, some folks will call it by a different name entirely