A name for "not quite saturated" graded modules

To elaborate on Karl's comment:

Let $m$ be the irrelevant ideal of $R$, then there is a short exact sequence:

$$0 \to H_m^0(M) \to M \to \Gamma^*(\mathcal{F}) \to H_m^1(M) \to 0$$

(see Eisenbud's book, Theorem A4.1, p. 693). Here $H_m^i(M)$ denote the local cohomology modules. So the map is injective precisely when $H_m^0(M):= \cup_n (0:_M m^n) = 0$. I believe such module is called $m$-torsion-free (don't know a reference off hand, may be Brodmann-Sharp's book on local cohomology?).

Also, it is equivalent to $m$ contains a non-zerodivisor on $M$ (may be that's what you meant in the last paragraph?)

As Hailong wrote, the injectivity means the vanishing of $H^0_m(M)$. But $\operatorname{depth}(m,M)$ is the least non-vanishing local cohomology, thus the map $M \to H^0_*(\tilde{M})$ is injective iff $M$ has depth $\ge$ 1. If $M$ has finite projective dimension, e.g. if R is regular, then this happens iff $M$ has projective dimension at most depth $R-1$ by the Auslander-Buchsbaum formula.