One is just the negative of the other, so up to a reflection the two definitions produce the same set.
Ziegler also defines a polytope as the set of all x such that Ax<=b for some matrix A and right hand side b. polymake uses 0\le b+Ax. Again, that just differs by a reflection in the origin. For both definitions you will find many books using them. Mathematically this is just a matter of taste (but you should be ocnsistent, if you use Ziegler's definition for a polytope, you should also use his definition of the polar, and vice versa).
For polymake there is actually a reason to use the definition it uses. polymake doesn't really work with polytopes, it operates on the homogenization cone of the polytope, i.e. if your polytope is the convex hull of v1, ..., vk, then polymake looks at the cone spanned by (1,v1), ..., (1,vk). See
here. In our representation, if 0<= b+Ax is an exterior description of your polytope, then 0<=bx0+Ax is one for the homogenization of P (in coordinates (x0,x). This makes the transfer fomr polytope to cone quite convenient.
Also, with our representation, if 0<=b_i+<a_i,x> is a facet defining inequality of P, then 1/b_i*a_i is the corresponding vertex of the polar, and if v_i is a vertex of P, then 0<=1+<v_i,x> is the corresponding facet of the polar. Thus, polarization in polymake is easy: Just exchange the matrices stored in FACETS and VERTICES (and scale each row such that the first entry is 1).
Hope that clarifies this. Feel free to ask again if not...
Andreas