Two definitions of smooth manifolds In Milnor/Stasheff they give the definition of smooth manifold as follows (page 4):
A subset $M \subset \mathbb R^A$ is a smooth manifold of dimension $n \ge 0$ if, for each $x \in M$ there exists a smooth function 
$$ h: U \to \mathbb R^A$$
defined on an open set $U \subset \mathbb R^n$ such that
1) $h$ maps $U$ homeomorphically onto an open neighborhood $V$ of $x$ in $M$ and
2) for each $u \in U$ the matrix $[\partial h_\alpha (u) / \partial u_j]$ has rank $n$.
This differs from the definition I know: $M$ is a smooth $n$-manifold if for any two charts $\varphi, \psi$ the transition map $\varphi \circ \psi^{-1}$ is smooth.
Are these two definitions equivalent?
 A: The definition in Milnor and Stasheff is a bit of a hybrid between a purely "coordinate chart" definition you alluded to (requiring transition functions to be smooth), and a purely Euclidean space definition, which runs as follows:
An $n$-dimensional manifold is a subset of $\mathbb{R}^A$ (here $A$ may be much bigger than $n$) such that each point has a neighborhood which is the graph of a differentiable function over a suitable coordinate subspace $\mathbb{R}^n\subset\mathbb{R}^A$.
Note that in this definition we only need the standard coordinate planes (it is not necessary to take arbitrary subspaces), by the implicit function theorem.
For example, the circle in the plane is the graph of a function of type $\sqrt{1-t^2}$ near every point, either over the $x$-axis or over the $y$-axis.
The "coordinate chart" definition has the advantage that no apriori structure is assumed on $M$ (other than being a set).  In particular, the topology results from the smooth structure imposed by the transition functions.
From this point of view, the Milnor-Stasheff definion has a disadvantage that we must already know about topological spaces and the notion of a homeomorphism.
