If it took a couple hundred years for this species to proliferate, it's absolutely plausible that eradicating it in a few years will have surprising second- and third-order effects.
Do you understand that time scales are important here?
There are also species that have invaded areas that are totally impossible for them to reach without seafaring or airfaring human civilizations to carry them.
All terms are made up. Some are useful and some are not. This one is useful.
Merely calling it invasive does not answer that question, ergo it is not a useful term in this conversation.
And yes, timescales do matter. A species can become critical to its new ("invaded") ecosystem within a few hundred years such that removing it within a few years will cause catastrophe.