Is a toric variety still a toric variety after change of coordiantes?

Question

I am reading Cox, Little, Shenck Toric Varieties, and I had a question about the definition of a toric variety. One of the examples (on page 12) uses $$x^2 - y^3 = 0$$ as an example of a toric variety. Is the following change of coordinates still a toric variety? $$(x-2)^2 - (y-2)^3 = 0$$ I feel it should be, but it seems from the comment about lattice points construction giving all the toric varieties (on page 14) that it is not attainable from the lattice points construction, since all the characters are maps into $\mathbb{C}^{\*}$, and there's no way to adjust the missing axes to line up with the cusp of the new curve. If it's not a toric variety, what part of the definition does it violate? It seems I can define a similar isomorphism from $\mathbb{C}^{\*}$ to $V((x-2)^2 - (y-2)^3) \setminus \{(2, 2)\}$, along with a similar action.