Definition of manifolds as submanifolds of $\mathbb{R}^m$

I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $$\mathbb{R}^m$$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).

The definition of a submanifold I learned is:

Let $$M$$ be an $$m$$-dimensional manifold. We say that a subset $$L \subseteq M$$ is a submanifold of $$M$$ of dimension $$n$$, if for every point $$p \in L$$ there exists an adapted chart $$\phi: U \rightarrow V' \times V''$$ with $$U \subseteq L$$ open in $$L$$, $$V' \subseteq \mathbb{R}^n$$ open in $$\mathbb{R}^n$$ and $$V'' \subseteq \mathbb{R}^{m-n}$$ open in $$\mathbb{R}^{m-n}$$ such that $$\phi (U \cap L) = V' \times \{0\}$$ with $$0 \in \mathbb{R}^{m-n}$$ Then $$L$$ becomes an $$n$$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $$\phi : U \cap L \rightarrow V'$$.

The authors of the book define a submanifold of $$\mathbb{R}^m$$ as follows

A subset $$L \subseteq \mathbb{R}^m$$ is called an $$n$$-dimensional submanifold of $$\mathbb{R}^m$$ if for every point $$p \in L$$ there exists an open set $$U \subseteq \mathbb{R}^m$$ containing $$p$$ and an open subset $$V\subseteq \mathbb{R}^m$$ together with a diffeomorphism $$\phi$$ from $$U$$ to $$V$$ such that $$\phi(M \cap U)=V \cap (\mathbb{R}^n \times \{0\})$$ with $$0 \in \mathbb{R}^{m-n}$$.

,which directly coincides with the book's definition if one views $$\mathbb{R}^m$$ as an $$m$$-dimensional manifold with the smooth atlas induced by the universal chart $$(\mathbb{R}^m, id)$$. Then the authors define coordinate charts of an $$n$$-dimensional submanifold $$L$$ of $$\mathbb{R}^m$$ around a point $$p \in L$$ as follows

Let $$\phi: U \rightarrow V$$ be a homeomorphism from an open set $$U \subseteq L$$ in the subspace topology containing $$p$$ to an open subset $$V$$ of $$\mathbb{R}^n$$ such that $$i_M \ \circ \ \phi ^{-1}$$ is a $$C^\infty$$-immersion, where $$i_M$$ is the cannonical injection from $$L$$ to $$\mathbb{R}^m$$.

How is this equivalent to the restricted charts given in the first definition? Why does $$i_M \ \circ \ \phi^{-1}$$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $$\mathbb{R}^m$$. What's a good way to think about the approach taken by the book?