I recommend reading up Amperes law especially the differential form
EM field theory is taught with the standardization of how Positive (commonly) charges are effected in a given scenario. The induced magnetic field due to a current carrying wire depends not only the charge of the current (positive vs negative) but also the charge you use as a reference to observe the effects of a magnetic field. If you have a wire with a current of negative charge and utilize a positive charge particle for observation, you will see it getting deflected in the counter clockwise direction. However, if you have a wire with a current of negative charge, but put a negative test particle in the region of the induced magnetic field, it would be clockwise rotation just as you saw with a positive charge current and positive test particle. Positive charge test particles are the usual convention. I hope I didn't miss your true question.
Once you take a look at amperes law, you'll quickly see that the dependence is on the current density (velocity and density of charges). The wire will effect the mobility of charged particles and depending on where you're measuring your magnetic field, the material can effect the magnetic field as well (magnetic permeability).