In the UK, August 12 is commonly known as ‘the Glorious Twelfth’ due to its being the start of grouse shooting season, which then leads on into other shooting seasons. This was as true in the Regency era as it is today in England, for August is when those who could—or who can today—fled London’s heat (and the stench of the Thames) for the countryside.
In Britain, ‘hunting’ is considered the proper term for hunting with hounds, as in a fox hunt. Deer hunting is more correctly called deer stalking, and ‘shooting’ applies specifically to shooting game birds. This is why you have George Underhill in his book A Century of English Fox Hunting noting: “In olden days it was the theory, and, I may add, the rule, that the months of September and October should be devoted to shooting, and that hunting commenced on the first Monday in November.” (Hunting meant fox hunting, and cub hunting would start up in late September or early October with drag hunts to train young fox hounds.)