I think there are over 100 species of chickweed. "Common chickweed" is most common (and exotic and invasive) species and is called
Stellaria media, while I was talking about "Star chickweed," which is
Stellaria pubera. Star chickweed is much, much less common (probably because it's not invasive!) and is a native of some parts of the US. We have some, but not very much. I was thinking about planting some more, if I can find some seeds! It sounds like the source for seeds you found sells Common chickweed seeds, if they were warning about it taking over.
You might have Common chickweed entrenched in your gardens! Here is a site that describes how to tell the difference between these two:
http://identifythatplant.com/stellaria-chickweed-species/