Are there only two irreducible affine singular cubic curves?

Question

It's known that every irreducible projective singular cubic curve in $\mathbb{CP}^2$ is projectively equivalent to either a cuspidal cubic $$y^2z=x^3,$$ or a nodal cubic $$y^2z=x^3+x^2z.$$ If we focus on the affine case, is it still true that every irreducible affine singular curve in $\mathbb{C}^2$ is isomorphic to either $$y^2 = x^3$$ or $$y^2=x^3+x^2?$$ If not, what is a counter-example?