Is the following definition of an elliptic curve correct? Im new to algebraic geometry so I want to make sure im getting my definitions right. I know there are a few ways to state what an elliptic curve is (ex a smooth projective curve of genus one with distinguished $K$-rational point). But I am wondering if the following is equivalent. For simplicity lets just work over $\mathbb{C}$.:
$\textbf{Definition:}$ An elliptic curve $E$ is a non-singular projective curve in $\mathbb{P}^2$ of the form
$$E: Y^2Z + a_1XYZ + a_3YZ^2 = X^3 + a_2X^2Z + a_4XZ^2 + a_6Z^3 $$
I am wondering if this is sufficient to define an elliptic curve?

Yes, this will do. The abstract definition of an elliptic curve is that it is nonsingular of genus $1$ as you mentioned. If your curve is cut out by a degree $d$ equation in $\Bbb{P}^2$, the degree-genus formula gives
$$
g=\frac{(d-1)(d-2)}{2}.
$$
Hence, since you have a cubic $C$, $g(C)=\frac{2}{2}=1$. Conversely, you can show that any abstract genus $1$ curve over an algebraically closed field $k$ has an embedding into projective space $\Bbb{P}^2$ which realizes it as a degree $3$ curve.
