Let $M$ be a finitely generated graded module over a graded ring $R$. Let $\mathcal{F}$ be the corresponding coherent sheaf on $\operatorname{Proj} R$. There is a natural map of graded $R$-modules
$$\phi \colon M \to \Gamma^*(\mathcal{F}) := \bigoplus_{n} \Gamma(\operatorname{Proj} R, \mathcal{F}(n)).$$
If I recall Ravi Vakil's notes correctly, $M$ is called *saturated* if $\phi$ is an isomorphism.

Is there a term (perhaps semi-saturated, or some such) for modules $M$ such that $\phi$ is injective?

This concept is appealing for several reasons. For one thing, it is easier to test "semi-saturatedness" than saturatedness; e.g., unless I am mistaken, $\phi$ is automatically injective if $M$ admits any positive-degree homogeneous nonzerodivisor. For another, at least if $R$ is a polynomial ring, $M = R/I$ is "semi-saturated" iff $I$ is a saturated ideal of $R$. (Note that the definition of "saturated ideal" is different from the definition given above for "saturated module", and I do not think the two are equivalent for ideals.)