The way to tell the last two apart is that, these days, "invective" is almost always used as a noun (as it is in the definition you cited). In older days it apparently was used as an adjective.
"Vituperative" is always an adjective. As commenters have noted, the noun form is "vituperation". One very rarely hears either spoken. Few Americans know these, and fewer still are willing to speak such a five-syllable word in public for fear of being thought erudite or even pedantic.
You could refer to a person as "abusive", but without further clarification that could mean physically abusive, verbally abusive or both. You could say he used abusive language. Anyone would know what you meant.
Invective is a good word to describe so-called rational arguments from supposedly intelligent people that are really disguised ad hominem remarks. William Buckley, for instance, was a master of this, disguising truly offensive views in flowery sesquipedalianism.