The definition of homogenous (like everything really) is different depending on who is defining it. In many mathematical contexts it refers to something involving a zero but I'm not sure what it means for your situation.
The norm of a vector should have the same definition for every vector in that space.
I am taking a look at this:
Homogeneous coordinates - Wikipedia, the free encyclopedia
and it says that homogenous co-ordinates refer to co-ordinates based on a projective space.
In a projective space, you don't have the same norm that you do in a Cartesian space: they are different kinds of spaces and subsequently they have different kinds of metrics, norms, and inner products (if these things exist for that space).