The first one and the third one are both okay as they are. The second one could also be okay if you added just a couple of words: The horse felt it when Rick pulled the reins.

Since others have answered your question, I will just comment on grammar. In #3, "pulling the reins" is an adjective clause that modfies "Rick". In #1, I believe "pull the reins" is an adjective clause that modifies "Rick". I find the grammar of #1 most interesting. Other similar sentences that use infinitives without "to" are: "I watched them dance" "I hear her sing" "You can relax" (this pattern is used with all the modals). "You can relax" is especially interesting. What is "relax"? You can dismiss the question by saying that, well, that's just the way that modals work but that would be a cop-out. I think "relax" is an infinitive behaving as a noun. It is a thing that you can do, hence a noun. What do you think?

The first one is the most natural. The third one is also correct.