Maybe I'm wrong, but I've researched this and looked at both your link and multiple other links. You actually mentioned it as well but I think you misunderstand what foreign means: "Foreign means that this entity is domiciled in another jurisdiction but registered/qualified to do business in Delaware."

This means in simple words the following: if a company (llc/corporation) is registered in Delaware, it is a domestic company. If a company is registered in a different state other than Delaware (say California for example) then it is considered a foreign entity and must be registered as a foreign entity in order to conduct business in Delaware. The same applies vice-versa; a domestic Delaware company must register as a foreign entity to do business in California for example.

Foreign company does not mean it's registered by foreigners. Nor does domestic mean it's registered by US residents. Either one can be opened by either US residents or non-US residents and neither is illegal.