Definately not, and I say that as someone who doesn’t even like anime. Anime is merely the style of art and has nothing to do with setting or theme or whether there’s sex or violence involved.
Anime just means “Japanese animation.” I suspect the additional information in the definition is just pointing out some of the common characteristics of Japanese Animation but not requiring it to have those.
Yeah, I don’t like either definition. Anime, as a Japanese word, just means animation. Thus anything animated, whether it be The Simpsons or Spirited Away, is anime. As an English word, anime means Japanese animation, but since animation isn’t a genre but a technique, anything animated and originally from Japan is generally called anime, even if it doesn’t fit the stereotypes.