modular forms and line bundle
Solution 1:
The statements are both true, but not because they are equivalent. The line bundle $L$ you consider is probably the dualizing sheaf $\omega$ (or just canonical sheaf or just sheaf of relative differentials depending on your base scheme). Then modular forms of weight $k$ correspond to global sections of $\omega$.
Maybe Peter Bruin's thesis could help here.
See http://user.math.uzh.ch/bruin/thesis.pdf
Especially
Equation I.2.1 (The language is a bit heavy in this section but it becomes more down to earth later).
page 76 might also be helpful.