Does centripetal force being equal to full tension dependent on if string is horizontal or vertical?

Firstly, the string cannot be horizontal, unless something supports the object.

Secondly, the centripetal force is not equal to the full tension in the string. The centripetal force acts in the horizontal plane the object travels in.

The forces:

$$Tcosalpha=F_c$$ $$Tsinalpha=mg$$ So that:

$$frac{mg}{F_c}=tanalpha$$

where $F_c=mromega^2$. You can see why $alpha=0$ is not possible: it requires $omega to +infty$!

So the centripetal force is proportional to the mass of the object.