The distinction to be made is that a differentiable structure is a choice of maximal smooth atlas $\mathcal A$, but two different choices $\mathcal A$ and $\mathcal A'$ can lead to isomorphic smooth structures. As an example, the canonical smooth structure $\mathcal A$ on $\mathbb R$ that contains the smooth function ${\rm id}:\mathbb R\longrightarrow \mathbb R$ is isomorphic to the smooth structure $\mathcal A'$ that contains the smooth function $x\mapsto x^3$, although $\mathcal A'\neq \mathcal A$. Thus, although a manifold admits uncountably many different smooth structures, it may have finitely many isomorphism classes of such structures.
Unfortunately Prof. Lee hasn't answered but I think I arrived at a satisfactory answer. There is a way to treat smooth manifolds and smooth manifolds with boundary simultaneously, however when the codomain is a smooth manifold the situation is slightly simpler.
General Definition. Let $M$, $N$ be smooth manifolds with boundary, $A\subseteq M$ and
$F:A\to M$ a map. We say that $F$ is smooth on $A$ if for each $p\in
A$ there is a neighborhood $W$ of $p$, a chart $(V,\psi)$ for $N$ and
a smooth map $\tilde F:W\to R^n$ such that $F(W\cap A)\subseteq V$ and
$\tilde F|W\cap A=\psi \circ F|W\cap A$.
Simplified Definition. If $N$ is a smooth manifold (without boundary) the general definition
simplifies to: We say that $F$ is smooth on $A$ if for each $p\in A$
there is a neighborhood $W$ of $p$ and a smooth map $\tilde F:W\to N$
such that $\tilde F|W\cap A=F|W\cap A$.
The following paragraphs justify the given definitions.
Let's prove that the general definition matches the simplified definition when $N$ is a smooth manifold (without boundary).
Simplified definition. $\implies$ General definition. Let $p\in A$, $\tilde W$ neighborhood of $p$ and $\tilde F:\tilde W\to N$ smooth such that $\tilde F|\tilde W\cap A=F|\tilde W\cap A$. Let $(V,\psi)$ be any chart for $N$ containing $f(p)$. Define $W=F^{-1}(V)$, then $\psi \circ \tilde F|W:W\to R^n$ is smooth because $\tilde F|W:W\to V$ and $\psi:V\to R^n$ are smooth. Also $F(W\cap A)=\tilde F(W\cap A)\subseteq V$. Finally $(\psi \circ \tilde F|W)|W\cap A=\psi \circ F|W\cap A$.
General definition. $\implies$ Simplified definition. Let $p\in A$, $W$ neighborhood of $p$, $(V,\psi)$ chart for $N$ and $\tilde F:W\to R^n$ smooth such that $F(W\cap A)\subseteq V$ and $\tilde F|W\cap A=\psi \circ F|W\cap A$. Define $\tilde W=\tilde F^{-1}(\psi(V))$, note that $\tilde W$ is a neighborhood of $p$ because $\tilde F(p)=\psi(F(p))\in \psi (V)$ and $\psi(V)$ is an open subset of $R^n$ because $N$ is a smooth manifold (without boundary). Now $\psi ^{-1}\circ \tilde F|\tilde W:\tilde W\to V\subseteq N$ is smooth because is the composition of $\psi^{-1}$ and $\tilde F|\tilde W$. Finally $(\psi^{-1}\circ \tilde F|\tilde W)|\tilde W\cap A=F|\tilde W\cap A$.
Let's prove that the general definition matches Definition 4(b).
Definition 4(b) $\implies$ General definition. Let $p\in A$, $\tilde W$ neighborhood of $p$ and $(V,\psi)$ chart for $N$ such that $F(\tilde W\cap A)\subseteq V$ and $\psi\circ F|\tilde W\cap A:\tilde W\cap A\to R^n$ is smooth as in Definition 3. Particularly there is a neighborhood $W$ of $p$ and a smooth map $\tilde F:W\to R^n$ with $W\subseteq \tilde W$ and $\tilde F|W\cap A=\psi \circ F|W\cap A$ ,i.e. pick $q=p$ in the notation of Definition 4(b). Finally $F(W\cap A)=\psi^{-1}(\tilde F(W\cap A))\subseteq \psi^{-1}(\psi(V))=V$.
General definition. $\implies$ Definition 4(b) Let $p\in A$, $W$ neighborhood of $p$, $(V,\psi)$ chart for $N$ and $\tilde F:W\to R^n$ smooth such that $F(W\cap A)\subseteq V$ and $\tilde F|W\cap A=\psi \circ F|W\cap A$. Let $q\in W\cap A$ (as in Definition 4(b)) and define $U=W$.
Best Answer
It is true that each manifold can be embedded into some Euclidean space $\mathbb R^N$. In other words, each manifold is dffeomorphic to a a submanifold of a Euclidean space, and therefore one could argue that it is sufficient to develop a theory for such submanifolds. In fact, some concepts (for example, the tangent space) even allow a more intuitive access for submanifolds than for "abstract" manifolds.
However, the embedding theorem is an existence theorem which does not provide a canonical embedding. There are many such embeddings, and each depends on certain choices. A priori it is not even clear what the minimal dimension $N$ of an ambient $\mathbb R^N$ is.
Many well-known manifolds are not given as submanifold, but by other constructions. Here are some examples:
Projective spaces $\mathbb RP^n$ and $\mathbb CP^n$
More generally quotients of manifolds by group actions.
Grassmann manifolds and Stiefel manifolds.
Quotients like $[0,1]/(0 \sim 1)$.
Try to find explicit embeddings of these objects into a Euclidean space - you will see it is not easy.
Thus I completely agree to the statement that