Strictly speaking, invasive means it can spread on its own and disrupt natural habitats, while noxious is a legal definition applied to plants that are invasive and that pose some kind of economic or public health hazard, usually stuff that's poisonous to either people or livestock (pets don't usually count on their own, not having a market value), or that causes economic harm to agriculture or forestry by reducing yields and requiring money be spent on control measures.

Whether something is invasive depends on the local environment. Broom is highly invasive here, but a treasured ornamental in the eastern US. Rhodies are scattered natives here, but Rh. ponticum, from Turkey, is a nasty invasive in UK forests. People I know who've seen it say "it's just like Scotch broom!" A rhodie hybridizer I asked about that said that non-native rhodies (and Japanese maples too) don't get established here as invasives because the seedlings can't survive our normal dry summers without irrigation. Once established they're drought tolerant enough, but it takes a few years.

So, dame's rocket. Yep it reseeds. So far it sticks to urbanized areas. That may or may not last. Sometimes a plant (or other critter) is restricted to localized, urbanized areas for decades, and not seen as posing a threat, but then it just explodes. Who knows why? probably different reasons in different cases. Weather changes, like a wet summer that doesn't kill the seedlings. Slow build-up of a sufficiently large population that then grows exponentially. Local evolution of the existing population to be better adapted to the new environment. Physical changes to the local environment that make it better suited, like clearcutting, urbanization, water diversions, plowing or bulldozing.

USDA lists dame's rocket as being noxious in Massachusetts, Connecticut and Colorado. UW describes it as 'sometimes escaping and persisting on waste ground'. So there's probably potential of it becoming a problem. I'm sure no one thought for a minute that Scotch broom would become the problem it is now. We just can't predict, and by the time we realize, it's too late to control the problem. The genie is out of the bottle.

Should you plant it? I dunno. I like it too, and love the fragrance. I don't have any but I'm tempted sometimes to gather seed from plants around the neighborhood. Yes apparently it survives Washington's summers without irrigation, but Oregon is drier, so I dunno. And I think foresters would dispute that foxglove does no harm. It fills in clearcuts, and takes up water and nutrients that could otherwise be used by regenerating native plants and the desired conifers. Water and nutrients are quite limited in much of our forest lands, and even something as seemingly minor as foxglove can tip the balance by taking up too much of something very limited. I've seen clearcuts in the western Olympics just solid with it. California poppies are a similar problem in our drier meadows and prairies, again where water and nutrients are limited and they spring up very early and take up too much of the limited supply. In urban areas neither one is a problem and could be considered urban beautification, but they don't stay in urban areas.