On pp.89 of these notes live TEX-ed by Arun Debray for Dan Freed's K-theory course there is a geommetrical definition of loop group of a compact Lie group $G$.
Let $P \to S^1$ be a principal $G$-bundle and $L_P{G}$ the group of automorphisms $\varphi$ of $P$ that cover $\text{id}_{S^1}$. If $P = S^1 \times G$ then it says that there is a very natural identification with "the" loop group: such an automorphism $\varphi$ is a loop through the action of $G$.
I am not quite sure of the last sentence, that such an automorphism $\varphi$ is a loop through the action of $G$ provides a natural identification of $L_P{G}$ with "the" loop group (and why double quote "the"?). Could somebody help to clarify?