In this context, a courser is a warhorse. The etymology was probably from the Italian 'corsiero' - 'battle horse'.
Coursers were used for hunting and for hard battle. While the destrier was a bit of a diva and most commonly used for jousts and single combat, the courser was much more like the modern cavalry horse. It was a lighter horse more suited to maneuverability on a battlefield.
The courser most closely resembled the modern hunters of northern Europe, but those lines almost certainly also went into the creation of the stock horse.
A "half-bred" hunter begs for a treat - although taller, this horse is probably not so very different from the medieval courser.