Bell got his .303 that he mentions in Karamojo Bell / Bell of Africa from Army & Navy Cop-Op, these were the same as the BSA catalogue I believe. He mentions it in one of his book or articles, Karamojo Bell, I think. It would have looked the same as the one pictured above, only it had a ten round magazine.
But his first .303 was a military Lee Metford in a sporter stock. He then mentions returning to Africa after that first safari, and bringing 2 x .303 Lee Enfield sporting rifles for his second expedition. He obviously used at least three maybe four over his elephant hunting years.