Consider an invertible sheaf $\mathscr L$ over a noetherian scheme $X$. I always knew that a local section of $\mathscr L$ is simply an element of $\mathscr L(U)$ whereas a global section, or simply a section, is an element of $\mathscr L(X)$.
Sometimes when I talk to people they say something like:
"a section is a morphism of sheaves $\mathscr O_X\to \mathscr L$"
Is it just a weird terminology or there is some kind of relatioships between the two concepts?
EDIT : I forgot the word "proper" in my example.
In this case the section people consider is the image of $1\in \mathcal{O}_X(X)$ which is an element of $\mathcal{L}(X)$.
This makes more sense when $X$ is a connected proper scheme over a field $k$ (e.g. a variety) : then as a consequence of Zariski's main theorem, $$\mathcal{O}_X(X)=H^0(X,\mathcal{O}_X)=k s_0$$
Then the image of $s_0$ is in $H^0(X,\mathcal{L})$.