In the literature, an elliptic curve $E/\mathbb{Q}$ is defined to be modular in two different ways
1) if there exists a nonconstant morphism $X_0(N) \to E$,
2) if there exists a modular form $f$ with $L(E,s) = L(f,s)$.
I get that from the first definition, one can deduce the second by pullback of a holomorphic differential on $E$ to a newform which has the same $L$-function, so my question is how does one get from the second to the first? Are both definitions equivalent? They seem to be used interchangeably by some authors but I have not been able to find any reference on this.
The book Diamond–Shurman "A first course in modular forms", in particular p. 362, tells you how $2)$ implies $1)$. This is actually a deep result: it requires to use Faltings' isogeny theorem.
If $L(f,s) = L(E,s)$ for some normalized newform $f \in S_2(\Gamma_0(N_E))$, then $a_p(f) = a_p(E)$ for every prime $p$ and then all the Fourier coefficients of $f$ at $i\infty$ are integers. In particular, the abelian variety $A_f$ constructed by Eichler and Shimura has dimension $[K_f : \Bbb Q] = 1$, where $K_f$ is the coefficient field of $f$.
Thus, we have an elliptic curve $A_f$ such that $L_p(A_f, s) = L_p(E,s)$ for almost every prime $p$ (equation 8.42 in Diamond–Shurman). Faltings' isogeny theorem asserts that you have an isogeny $\phi : A_f \to E$. On page 247 of Diamond–Shurman, using the decomposition of $J_0(N)$, one gets an isogeny $J_0(N) \to E$, and on page 216, we get a non-constant holomorphic map $X_0(N) \to E(\Bbb C)$. Finally, it is mentioned on page 292 that we get a non-constant morphism $X_0(N) \to E$ defined over $\Bbb Q$ (and work of Carayol ensures that we may take $N = N_E$).