Equivalent definitions of a surface do Carmo Differential Geometry of Curves and Surfaces defines a regular surface as per the below post.
Lee Introduction to Smooth Manifolds defines an embedded or regular surface to be an embedded or regular submanifold of $\mathbb{R}^3$ of codimension 1, namely a subset $S\subset\mathbb{R}^3$ that is itself a smooth $2$-dimensional manifold whose topology is the subspace topology and whose smooth structure ensures the inclusion map $\iota:S\hookrightarrow\mathbb{R}^3$ is an embedding.
Question:  Are these definitions equivalent?  If so can someone present or point to in the literature a detailed proof.

The answer to my question is yes, and is given by Theorem 5-2 in Spivak Calculus on Manifolds combined with Theorem 5.8 in Lee Introduction to Smooth Manifolds.  See also here.
