In the USA There is Executive Order 13112 which states:-
"Invasive species" means an alien species whose introduction does or is likely to cause economic or environmental harm or harm to human health.
In South Africa there is the law in the form of the National Environmental Management : Biodiversity Act, 2004 Alien and Invasive Species Regulations
The regulations use the following definition.:-
"invasion" means the establishment and subsequent spread of a species outside its distribution range in a manner or to the extent that it constitutes a threat to biodiversity.
The Australian Government defines the term as follows :-
An invasive species is a species occurring, as a result of human activities, beyond its accepted normal distribution and which threatens valued environmental, agricultural or other social resources by the damage it causes. 
and also like this:-
Invasive species are animals, plants, parasites or disease-causing organisms that establish outside their natural range and become pests. 
This is the usage found all across the English speaking world. This is an example from the government of the Philippines  :-
"Invasive species" means an alien species which becomes established in natural or semi-natural ecosystems or habitats, is an agent of change, and threatens native biological diversity.
It is important to know whether a plant is native or not because plants behave differently when they are in foreign places where the insects that eat them do not occur. Research shows that :-
in its native range, which extends from western Europe to Central Asia, ragwort is attacked by more than 70 herbivorous insects, several of which have a very narrow host-range. (Which means they only feed on ragwort or a few similar plants.) 
Research has shown  that ragwort, for example, becomes much more toxic when introduced to other places. This is what the research actually said:-
Populations in the introduced range had 1.5-fold higher concentrations of Pyrrolizidine Alkaloids and produced twice as many flowers than populations from the native range.
4. Harper JL, Wood WA (1957) Senecio jacobaea L. J Ecol 45:617-637
5. Carole Rapo C., Muller-Scharer H., Vrieling K, Schaffner U,Is there rapid evolutionary response in introduced populations of tansy ragwort, Jacobaea vulgaris, when exposed to biological control? Evol Ecol (2010) 24:1081–1099
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