Bundles that have many sections go under the name of very ample bundles.
Line bundles can be defined over any field $k$ but it has to be the same as the field of definition of your original manifold. Why the field cannot be different? By definition a line bundle is something whose total space is locally $A^1\times B$ for $B$ as base.
Now I am not sure about less ad hoc way. Of course, what you're doing can be written more formally and more invariantly. Take the global sections $V = H^0(L)$ for a line bundle $L$. By definition, for a given point $x$ we have a linear pairing $V \times \{x\} \to L_x$, the fiber of $L$ at $x$ (which is isomorphic to $k$). A linear map to a one-dimensional space (even though the isomorphism with $k$ is not canonical) by definition gives a canonical point in $\mathbb PV^*$.
As you vary $x$, you get the map to $\mathbb PH^0(L)$, a very well-defined and canonical projective space. As the intuition would tell you, for this to be effective there should be a way to tell two points apart, that is if I have points $x, y\in X$ then there should be a section $s$ such that $s(x) = 0$ but $s(y) \ne 0$.
It's hard to construct sections of an arbitrary line bundle, but for the curves situation is quite easy. Every line bundle there has the form $O(D)$ for some divisor $D$. As you add more points to your divisor, you're guaranteed lots of sections. Therefore, there are many very ample sheaves.