Why are hydrides of transition metals called hydrogen-deficient?
As you note, many hydride phases have the hydrogen occupying interstitial sites within a metal phase, not unlike many carbide/nitride/oxide phases as well. In theory this could range from no hydrogen at all up to full occupancy (and perhaps even double occupancy).
Since you mentioned lanthanum hydride, I pulled up a Calphad analysis for the H-La-Ni system, focusing on the H-La part. Hydrogen has almost no solubility in dhcp-La, yet shows a wide solubility range in both fcc-La and bcc-La phases.
Now for the hydride phase, where I'll quote from section 3.4 in the paper above.

The binary LaH$_{x}$ hydride has an fcc structure... Usually, La atoms occupy the 4a sites, while hydrogen atoms occupy the tetrahedral 8c and octahedral 4b sites. Experimental determinations using using several different techniques ... show that hydrogen atoms occupy tetrahedral sites first, for concentration up to $H/M = 1.95$, and only above this concentration value octahedral sites begin to be occupied.

Their calculated diagram is:

So, hydrogen happily fills in to interstitial spaces, but prefers some more than others. Eventually (perhaps strain, perhaps getting too close to each other) other sites become more desirable for the additional hydrogen. In any event, often a quite broad range of compositions are stable, and perhaps (depending on the metal) you never make it to the ideal stoichiometric composition.