Organisms being farmed by ants, bugs latching onto a bird and travelling far away, seeds getting stuck in animals’ furs and being spread - these are all natural ‘wild’ processes. Even humans have been part of this natural process for ages, before we had bikes and cars and trains and airplanes that could bring us (and our ‘guests’) wherever we want within unnaturally short periods of time.
However, I feel humans have greatly transcended their original place in nature and now we are, somehow, too disconnected from nature to be considered a part of it - or at least a part of the slow natural processes.
Sure, some seeds or a tick that got stuck on you while you were walking the dog nearby, then got unstuck in your garden and propagate there are completely wild to me. Even a random plant that popped up in your garden and stays alive because you water it, is wild to me. And so is a caterpillar that you found in your garden, even though you took it inside and are actively caring for it. Whether you interact with it or not does not change anything to its wild origins. Now if you start a breeding program, carefully selecting the best seeds or the strongest butterflies, the offspring should be considered cultivated, as you’re actively changing the course of evolution.
However, some seeds that got stuck inside your shoes while you were travelling in California, then get planted unintentionally in your garden in New York - those plants are definitely not wild. They would have never gotten from California to New York within a day by any natural process. They were introduced by you, even if it was unintentionally. Same goes for a caterpillar that chooses your tent in Italy to pupate on, then emerges as adult moth in the UK. That moth is not a wild UK species and would have never reached the UK by any natural process, but could only get there because you brought it there by car or airplane.
I guess what I’m trying to say comes down to this: If you move an organism (intentionally or unintentionally) to a new place that it could have reached by any other natural process anyway, it should be considered wild. If it could not, it should be considered introduced (not wild) to the new spot.
A single introduced organism could probably not propagate, as there are no other individuals of the same species present. However, if more individuals are introduced and they start propagating, they could become an invasive species, which is dangerous for the local native biodiversity - which is why it’s best not to move organisms from one place to another.