In Us English, we don't call them "presenters". As Jerome said, we'd use "personality" or "host". And we usually use "show" rather than "program". I think your terms are fine in UK English.
However, I'd write this a little differently because "run" isn't quite the right word. Also, by "devoted" do you mean the animals are performers or are shown in nature?
He is famous for hosting TV shows featuring animal acts.
He is famous for hosting TV shows featuring animals in their native habitats.