There are several tiers of nonnative species commonly accepted in invasion ecology.
present- species has been sighted not much other evidence
Reproducing- population shows signs of reproduction may or may not be a permanent population
Established- population has persisted for multiple years and major swings in environmental conditions (think 10year record heat or cold for the area floods or droughts)
Invasive- established species that causes harm to the environment, humans, animals, plants, or economy
The other thing we need to discuss is the likelihood of a species moving from its original established range (range expansion) Peacock bass in a power plant cooling pond in Montana are never going to move beyond that pond. Bighead carp in the Mississippi are going to (have) travel to many of the tributaries up and downstream from their introduction point.
Then you have alternate ends for establishment.
Failed- the population was unable to persist (increase in natural predation, extreme weather event, etc wipes out population
Extirpated- human intervention removes the population to the point of failure
As for moving within a country's borders that is complicated and more political than ecological. Most people don't stress about an animal moving from one body of water to the next, but if it didn't get there naturally it really should be considered nonnative. As for invasive see the previous definitions. Now with natives we also have to talk about natural range expansions. These are species that aren't commonly found in an area but have moved there on their own, and keep expanding.
This is an extremely important conversation to have as detractors to our hobby want to say all of our animals are invasive or potentially invasive. We have to be vigilant to keep this from being politicized and call out bad science when we see it. A great example of bad science is here in the USGS database which is usually good.
Raphael catfish (Platydoras costatus) - Collection record (usgs.gov) . If you look at their collection record, they say they found one fish and listed it as established in Texas. These aren't self-cloning crawfish it takes 2.