Your definition (if made precise) is quite long, and it does not worth to define polyhedral space this way. 

You may do the following two things:

 1. Define it as a simplicial complex with a length metric such that each simplex is isometric to Euclidean simplex.

 2. Define it as a compact (or locally compact) metric space such that each vertex adimits a cone neighborhood. 

The second definition is weaker than yours and the first is stronger.
The equivalence of these two definitions (at least in compact case) is written, 
see  Proposition 2.12 [Lebedeva's paper][1]. 

  [1]: http://arxiv.org/pdf/1111.7253.pdf