Two or more sources of study I use on Differential Geometry,most notably, Barrett O'Neill's book, defines an $ n $ dimensional manifold as follows: A set furnishes with a collection $ P $ of abstract "patches" (which are 1-1 and regular functions x $ :D \rightarrow M $, $D$ being an open set in $ \mathbb R^n $) satisfying : The covering property: Images of patches in $ P $ cover $M$.
The smooth overlap property:For an $x$,$y$ in $P$, functions $y^{-1}x$ and $ x^{-1}y $ are differentiable and defined on open sets in $\mathbb R^n$ .
The Hausdorff property.
I would like to know if this definition is any better than the manifold definition involving charts that one normally encounters in which the above properties are stated for $x^{-1}$ . It is to be noted that these discussions are all mainly concerned with $2-D$ surfaces. So is this definition merely well-suited because it helps the fresher realise that surfaces in $3-D$ space are stitched out of $2-D$ subsets of the plane??
One another thing is that regularity and 1-1 avoids self-intersecting surfaces and other problematic ones. So is this definition any good than the atlas-based one??
Is there anything more subtle that I am failing to grasp here, or is it merely a matter of convention, in which case this question would probably look foolish?