The Sentinel Idol “Inspiration on Kills with Void Skills” does only work with skills from the Void Knight tree and not with skills that have the void tag.
I tested this with:
1.) Hammer Throw - Disintegrating Aura (skilltree node),
2.) Smite - Temporal Corruption (Passive in Void Knight)
3.) Abyssal Echo
1, 2 and 3 all have the “void” skill tag but only 3 could trigger the Inspiration buff from the idol.
One might think that this is actually intendend. However, this is inconsistent because the wording “something with [tag] skills” which is used quite often and always means the “tag” of the skill.
For example the unique Soulfire says Chance on hit to ignite with Fire Skills and Necrotic Skills and - of course - this works with every skill that has the fire or necrotic tag - no matter from which class-line it is.