If I look at the Wikipedia one-line definition again, that seems to match:

nihilism: a viewpoint that traditional values and beliefs are unfounded

...a sensible move away from religious, traditional values...

and that existence is senseless and useless.

...which is branded by religionists as leading to thinking "existence is senseless and useless", although that's both empirically and logically wrong. This part is the 'meaning' of 'nihilism' in the vernacular, as you say.