On page $30$ in Hirsch's Differential Topology one can find the sentence
These are images of embeddings, and should be "submanifolds".
This would imply that the definition of $C^r$-submanifold of a $C^r$-manifold $M$ (possibly with boundary) should be that it is a subset $N \subseteq M$, such that there is a $C^r$-embedding $f: N' \hookrightarrow M$ with $N = f(N')$.
But in the following paragraph, submanifold is defined to be a subset $N \subseteq M$ such that for every point $x \in N$ there is a chart $\varphi: U \to \mathbb{R}^n$ such that $\varphi(N \cap U) = H\times \{0\}$ for some $k$-halfspace $H \subseteq \mathbb{R}^k$.
With the above embedding definition the subset $\partial B((r,0),r) \subseteq \bar B(0,2r)$ ($B(x,r) \subseteq \mathbb{R}^n$ ball with radius $r$ about $x$) should be a submanifold of $\bar B(0,2r)$ but with the other definition not. So, these definitions are contradicting each other. What is the correct definition for submanifold of a $C^r$-manifold (with boundary; if there is no boundary both definitions coincide by Theorem $3.1$ Chapter $1$ in Hirsch, Differential Topology)?
Theorem $3.1$ is:
Let $N$ be a $C^r$-manifold, $r \geq 1$. A subset $A \subseteq N$ is a $C^r$-submanifold if and only if $A$ is the image of a $C^r$-embedding.
Thus, more precisely, the analog of this theorem for $C^r$-manifolds with boundary would not hold, if one defines submanifold as in the second defintion above?