I want to ask a related but distinct question to this post here about viewing manifolds as embedded subsets of $\mathbb{R}^n$.
I notice that in a lot of classic texts on differential geometry/topology that manifolds are usually defined in terms of possessing local charts diffeomorphic to $\mathbb{R}^k$, but under the general assumption that the manifold is already embedded into $\mathbb{R}^n$ for some $n\geq k$. This is what I've seen in a lot of John Milnor's work (Topology from the Differential Point of View and Characteristic Classes) as well Guillemin and Pollack's Differential Topology. However, more modern texts on the subject (e.g. Lee) define manifolds in terms of a set without reference to any overarching space in which it is already embedded.
My question is: why did authors choose to present manifolds in terms of their embeddings in $\mathbb{R}^n$ back in the 50s, 60s, and 70s, as opposed to their atlas constructions in the modern era?
I'd like to know if this was a matter of choice or style or if the atlas construction of manifolds was generally not used at this time. Obviously, any manifold defined in terms of an atlas construction can be embedded using one of the embedding theorems (Nash, Whitney). One advantage I can see (as pointed out in the cited question at the beginning of this post) is that one immediately has the notion of a chart being a diffeomorphism if we assume the manifold is already embedded in Euclidean space. This is in contrast to the atlas construction where the chart representations are assumed to only be homeomorphisms, but the transition maps between charts are imposed to be diffeomorphisms.