Consider the following definition of submanifold:
1.5. $\ \bf Definition.\ $ A subset $M\subset\mathbf R^{n}$ is called a $\underline{\text{differentiable submanifold}}$ of $\mathbf R^n$ of $\underline{\text{dimension}}$ $m\leqslant n$, if to each $x\in M$ there corresponds an invertible germ $\widetilde{\phi}\colon (\mathbf R^n,x)\to(\mathbf R^n,0)$, such that $\widetilde{\phi}(M,x)=(\mathbf R^m,x)\subset(\mathbf R^n,x)$ ($\mathbf R^m$ linearly imbedded in $\mathbf R^n$ for $m\leqslant n$)
It seems to me that this is not equivalent to the usual definition of a submanifold: the usual definition is that $M$ is a submanifold of $N$ if there is a map $f: M \to N$ such that $f$ is a smooth injective embedding that is a homeomorphism onto its image.
As far as I can tell the definition above merely says that $M$ comes with an atlas consisting of smooth diffeomorphisms $\phi$. But then any subset $M$ of $\mathbb R^n$ that can be endowed with such an atlas would be a submanifold of $\mathbb R^n$.
Am I missing something? Please could someone clarify this to me?
There are two main kinds of submanifolds:
embedded submanifolds (also called regular) have subspace topology. Appear as images of smooth embeddings. Charts on these submanifolds can be made by setting some unused coordinates to constants.
immersed submanifolds do not necessarily have the subspace topology. Appear as images of smooth immersions. For example, the dense curve on torus. There is no way the chart on the dense curve can be obtained by restricting a chart on the torus.
See submanifold article in Wikipedia, or much better, see Chapters 4 and 5 of Lee's Introduction to Smooth Manifolds. I think the definition you quote amounts to the embedded case. So, I would disagree that $M$ is so free from the topology of the ambient space. In particular, see Theorem 5.8 on page 101 of Lee's text where he explains that the $k$-slice condition implies the structure of an smoothly embedded submanifold.