Definition of an elliptic curve $E$ over $\Bbb{C}$ is defined over a subfield $⊆ℂ$

Question

An elliptic curve $E$ over $\Bbb{C}$) is defined over a subfield $⊆ℂ$

1. if $E$ is isomorphic(over $\Bbb{C}$ to $E_0$ whose coefficients are in $K$

2. if there exists an elliptic curve $E_0$ given by a Weierstrass equation with coefficients in such that $E\cong E_0×_{K}{Spec(\Bbb{C})}$ as curves over $ \Bbb{C}$.

I encountered two above seemingly different definition. Are these equivalent condition ? If so, what is the metric of bringing out fiber product-like definition ?