An elliptic curve over a field $K$ is defined as a 'genus $1$ curve with a base point'.
My question is, why don't we adopt the definition 'a smooth curve that is isomorphic to a degree $3$ smooth curve with a base point'?
I think these definitions are equivalent because a genus $1$ curve with a base point is isomorphic to the Weierstrass form over $K$, which is a degree $3$ smooth curve with a base point. On the other hand, every smooth curve that is isomorphic to a degree $3$ curve has a genus $\frac{(3-1)(3-2)}{2} = 1$.
What is the merit of defining elliptic curves as 'a genus $1$ curve with a base point' rather than as 'a smooth curve that is isomorphic to a degree $3$ smooth curve with a base point'?