> not all weeds are invasive (i.e. non indiginous)
I believe you are mistaken or confused. You also misspelled indigenous.
Invasive typically refers to the propensity of a non indigenous species (~= "plant that comes from somewhere else") to spread and replace or out-compete for resources such as light, nutrients, water, etc. other species that are indigenous species. At least in my travels, it is exceptionally rarely used in a broader sense to mean "any non-indigenous plant" (as adopting this sense would, for example, include pretty much every single garden plant ever, every food crop ever, etc. and thus be of dubious to zero utility).
An endemic (~="was always here and here only") species is not a weed, to a botanist, by definition, because it was always there and there only.
An indigenous (~="was always there") species is not a weed, to a botanist, by definition, because it was always there.
We often substitute the word "native" for indigenous in Australia and New Zealand, but in Australia this has an alternate meaning which is "comes from anywhere in Australia (not necessarily this specific area)" and therefore in an Australian context a "native" can be non-endemic, non-indigenous, and even invasive. An example of an invasive native would be Acacia longifolia when found in its non-indigenous range in Western Australia.
A weed in common use as a botanical pronoun by the general public most usually refers to any plant which is both invasive and non-indigenous. Any plant which is non-indigenous (but not significantly invasive) may also often be referred to as a weed by environmentalists, ecologists, and botanists (or anyone else), depending upon their mood, company and character.
I hope this helps clear up any confusion.
Sincerely, card carrying member, Australian Plant Society.