Here's my humble opinion on the subject, for what it's worth: If we go by a federal list of what is invasive and noxious, we won't be doing that much good.
I've been a terrestrial gardener for years and have seen this same question come up on plant trading forums again and again. The problem is our country is so vast and varied that what's invasive in one area isn't in another. To make sure they list ALL invasives, the list would include just about every plant we grow regularly except for true natives. They did kick around an idea to try that years ago ~ Google up "White list". They soon found out that wouldn't really be practical and would just about be impossible. The list they have now isn't really practical either, or accurate across the country. States even have trouble with coming up with an accurate list of all noxious weeds.
For instance, regular water hyacinth is one of the plants listed as illegal in my state, but I can't keep it alive for the life of me, even with a greenhouse to keep it in in winter. It's illegal in my state because it can get out of hand way down south along the coast, so rather than make it illegal just in that area and have to list the counties, they made it illegal all over the state, including up here where I am in Central Texas despite it having very little chance of surviving much less thriving and becoming a problem. Same with some of the salvinia species and others on the state list.
The same thing holds true for the federal list ~ some will be invasive in temperate areas such as Florida, while those same ones won't even last overnight in an Oklahoma winter. From the list above, azolla, sagittaria, and ipomoea are three more that won't even last through one of my mild Central Texas winters, atleast not in the sheltered valley I live in.
Then there are the truly noxious weeds that aren't on the list, but are noxious just about everywhere I've ever heard of them being grown. Kudzu and Johnson grass are two of those ~ yet they're not on the federal list (atleast they weren't last time I checked, a few years ago). A funny note is that the federal government is the entire reason they're here ~ they planted them on purpose long ago for soil erosion control, not realizing the consequences. A few others that aren't on the federal list, but are invasive in more places than plants that are on the list ~ China Berry trees, mimosa trees, taro, wax leaf ligustrum, butterfly bushes, blue fescue grass, and bermuda grass. Next time y'all go to your local Home Depot or Super Wal Mart, check how many of those are being sold in their garden centers. I'd bet my ranch half of them will be there.
What I'm saying, again, is that if we go by a federal list of what is invasive and noxious, we won't be really doing that much good. Yes, it's VERY good to know what you have and treat it responsibly. Most definitely!! But I believe we should do that with EVERY single plant and animal we keep, not just the ones on a noxious weed list. Many plants and animals we keep haven't been around long enough for anyone to know if they're invasive or not. It takes time to find that out. So if we don't put our faith in any list, but instead simply act responsibly with EVERYTHING we keep, we won't be the ones that unknowingly cause an ecological disaster.