Difficulties on Understanding different Definitions of a curve

Question

I do not really understand the connection between these two definitions: A regular simple (smooth) closed curve is a $ C^1 $-map $ \gamma:[0,1] \to \mathbb{R}^3 $ with $ \gamma^k(0) = \gamma^k(1) $, k=0,1, $ \gamma'(t) \neq 0$ for all $ t \in [0,1] $ and $\gamma|_{[0,1)} $ is injective. The other definition I encountered is the following: A regular (immersed) simple closed curve is an embedding $ \tilde{\gamma} : S^1 \to R^3 $ with $ \tilde{\gamma}'(t) \neq 0 $ for all $t \in S^1$. I see that there might be an equivalency but how would one write down the connection between these two definitions? I already found a result that states that we can " identify " $ [0,1]$ with $ S^1 $ in the case of closed curves. But how can I derive the other properties? For example, why would be $ \tilde{\gamma} $ be regular if the other representation $ \gamma $ of the curve is regular?