Let $\text{Emb}(S^1, \mathcal{I})$ denote the space of all embeddings of $S^1$ into $\mathcal{I}$. Each embedding $f_\gamma: S^1 \to \mathcal{I}$ is a smooth, injective map that traces out a simple closed loop on $\mathcal{I}$. Two embeddings $f_1, f_2: S^1 \to \mathcal{I}$ are considered equivalent if they are differentiable isotopies, meaning there exists a smooth family of embeddings connecting $f_1$ to $f_2$.

Would you say that this is well-written specifically in terms of good notation and modern standards?

Two remarks:
1. Axioms for many algebraic structures like groups or rings can be formulated 'internally' in many categories in a diagrammatic fashion. This gives rise to notions like group objects, ring objects, etc. in a category. This can be done with any sort of algebraic structure in an appropriate universal-algebraic sense.
2. Your examples do not obey these axioms, as you note, but only obey these axioms 'up to coherent isomorphism'. So what you're really looking at is a higher-categorical version of an <algebraic structure> object (in a higher category of categories).