6 years later Wiki seems to clear the subject:
Venison is the culinary name for meat from the family Cervidae. Deer meat, whether hunted or farmed, is termed venison.
Venison can describe meat of any mammal killed by hunting. It was originally applied to any animal from the families Cervidae (deer), Leporidae (hares), and Suidae (wild pigs), and certain species of the genus Capra (goats and antelopes), such as elk, red deer, fallow deer, roe deer, moose, reindeer/caribou, pronghorn, brown hare, arctic hare, blue hare, wild boar, and ibex, but its usage is now almost entirely restricted to the flesh of various species of deer. In Southern Africa venison is the meat of antelope. There are no native Cervidae in Africa.
so in Leo should be given both meanings: one historical and one modern.