I think my confusion has been cleared now, and finally understand what you've beeing trying to tell me (Sorry for being so slow!). If it's okay I will add what I understood also as an answer in case someone gets confused the same way as I did, please tell me if I write further nonsense.

My faith is now restored in the following statement:

Unless $-1 \in \Gamma$ and $k$ is odd, there exists an invertible sheaf $\mathcal{L}_k$ on $X(\Gamma)$ such that modular forms of weight $k$ for $\Gamma$ are in 1-to-1 correspondence with the global sections of $\mathcal{L}_k$.

Suppose $\Gamma' \subset \Gamma$ is a small normal congruence subgroup, then there is a very nice invertible sheaf $\omega$ on $X(\Gamma')$ and modular forms of weight $k$ for $\Gamma'$ are global sections of $\omega^{\otimes k}$. Let $\pi : X(\Gamma') \rightarrow X(\Gamma)$ be the projection. In Diamond and Im, the sheaf $\mathcal{G}_k$ is constructed as $\pi_{\ast}(\omega^{\otimes k})^G$. I still don't know if this sheaf is always invertible (perhaps it is). But I think it doesn't matter whether it is or not. It is coherent of generic rank 1 with possible torsion supported only at the elliptic points, so one can just define $\mathcal{L}_k$ to be the locally free part of $\mathcal{G}_k$.

My confusion came from this phenomenon: the identification of modular forms with global sections of $\mathcal{L}_k$ is unnatural. On $X(\Gamma')$, the global sections of $\omega^{\otimes k}$ are really modular forms, so $E_6$ as a section of $\omega^{\otimes 6}$ really vanishes at the orbit $[i]$.

On the other hand, the global sections of $\mathcal{L}_k$ correspond to modular forms, but they're not really modular forms (not in the sense I had in mind). Of course, this is due to the fact that $\pi^{\ast} \mathcal{L}_k$ is in general not isomorphic to $\omega^{\otimes k}$. Thus on $X(SL_2(\mathbb{Z}))$, $E_6$ as a section of $\mathcal{L}_6$ does not vanish at $[i]$. So the invertible sheaf $\mathcal{L}_6$ is actually trivial for having a nowhere vanishing section. (This explains why its space of global sections is 1-dimensional.) My argument above showing that $E_6$ vanishes, of course, shows that it vanishes on the moduli stack.

Thanks eric and David for helping me out!

I think my confusion has been cleared now, and finally understand what you've beeing trying to tell me (Sorry for being so slow!). If it's okay I will add what I understood also as an answer in case someone gets confused the same way as I did, please tell me if I write further nonsense.

My faith is now restored in the following statement:

Unless $-1 \in \Gamma$ and $k$ is odd, there exists an invertible sheaf $\mathcal{L}_k$ on $X(\Gamma)$ such that modular forms of weight $k$ for $\Gamma$ are in 1-to-1 correspondence with the global sections of $\mathcal{L}_k$.

Suppose $\Gamma' \subset \Gamma$ is a small normal congruence subgroup, then there is a very nice invertible sheaf $\omega$ on $X(\Gamma')$ and modular forms of weight $k$ for $\Gamma'$ are global sections of $\omega^{\otimes k}$. Let $\pi : X(\Gamma') \rightarrow X(\Gamma)$ be the projection. In Diamond and Im, the sheaf $\mathcal{G}_k$ is constructed as $\pi_{\ast}(\omega^{\otimes k})^G$. I think it doesn't matter whether it is invertible or not (probably it is). It is coherent of generic rank 1 with possible torsion supported only at the elliptic points, so one can just define $\mathcal{L}_k$ to be the locally free part of $\mathcal{G}_k$.

My confusion came from this phenomenon: the identification of modular forms with global sections of $\mathcal{L}_k$ is unnatural. On $X(\Gamma')$, the global sections of $\omega^{\otimes k}$ are really modular forms, so $E_6$ as a section of $\omega^{\otimes 6}$ really vanishes at the orbit $[i]$.

On the other hand, the global sections of $\mathcal{L}_k$ correspond to modular forms, but they're not really modular forms (not in the sense I had in mind, functions on the moduli space of elliptic curves). Of course, this is due to the fact that $\pi^{\ast} \mathcal{L}_k$ is in general not isomorphic to $\omega^{\otimes k}$. Thus on $X(SL_2(\mathbb{Z}))$, $E_6$ as a section of $\mathcal{L}_6$ does not vanish at $[i]$ (as David said, $E_6$ is the local basis). So the invertible sheaf $\mathcal{L}_6$ is actually trivial for having a nowhere vanishing section. (This explains why its space of global sections is 1-dimensional.)

Thanks eric and David for helping me out!