I have been studying group actions for about a month, so I am at a basic stage and forgive me if this turns out to be a stupid question. It is about the two equivalent definitions of group action and permutation representation. In the 'action' definition of a left group action, we use a map $ \alpha: G \times X \rightarrow X $, which I understand can be curried to give a chain of two functions $ \phi : G \rightarrow (X \rightarrow X) $, which leads to the alternative definition of the group action as a permutation representation $ \phi : G \rightarrow Sym(X)$.
But how does this work for the right group action $ \alpha: X \times G \rightarrow X $, as following the same currying technique would give a chain of functions $ \phi: X \rightarrow (G \rightarrow X)$ and not a permutation representation?
In addition, what are the main reasons for wanting two equivalent definitions as an action and permutation representation? It seems to me that the definition as a group representation is quite natural, whereas the action definition is axiomatic and seems to come from nowhere. Many thanks.