Is it "finger clicking"? What verb should I use if not "click"? "Snap"? Are both ok? It's just that if I say "finger snapping", one may imagine something violent.
Yes "snap your fingers" is the common phrase, but "clicking" is also an alternative. The image you searched was even labelled as "finger snapping"!
Snapping (or clicking) one's fingers is the act of creating a snapping or clicking sound with one's fingers. Primarily this is done by building tension between the thumb and another (middle, index, or ring) finger and then moving the other finger forcefully downward, so it hits the palm of the same hand at a high speed. Wikipedia
to snap one's fingers is the common phrase. It has nothing to do with digital mayhem; rather it means to produce a snap sound with the fingers, as if by breaking a twig.
I think the actual sound is produced by one of the fingers slapping against the meaty part of the palm under the thumb after the finger is held under tension and then suddenly released.