What is wrong with this defintion? Nothing really, it is just that it is a bit of a pain to have to construct an `extension' of X across the boundary to show that it is a manifold with corners. For this reason I want to give a more intrinsic definition (which gets me into a bit of trouble) and then leave it to you to check some of the details showing that the same spaces arise from the two approaches, that is to show that this is an alternative characterization of manifolds with corners.
Using the open subsets of 
as the model spaces and the
defintiion of a manifold (without boundary) as the starting point it is
natural to try
Definition A tied manifold is a paracompact, Hausdorff
topological space X which has a covering by open subsets 
with homeomorphisms onto open sets 
such that for each pair 
for
which 
the map
has the property that
First notice that this all makes sense. The sets
are open, so we have already defined
the spaces 
The main condition here
just says that if 
then 
Of course you should not expect this definition to give exactly the same
spaces as the previous one, otherwise I would not have wasted time giving
them a different name! To see what the relationship is precisely, first we
can define 
for any tied manifold by
We can then say that a manifold with corners is a tied manifold, meaning
that its space arises this way. Not only can we
say it, but it is true.
Theorem Any manifold with corners is a tied manifold. A tied
manifold is a manifold with corners if and only if there is a finite
collection of elements 
such that if
and 
is such that 