Were the Puritans here "legally"? According to whom? What about the Spaniards in Florida? The State of Texas? The Kingdom of Hawaii?

is basically "someone who is in country legally"

H1B's are Americans? (I don't agree with that statement)

None of this matters, btw. As I pointed out, there are those who blame Liberals/Democrats for ignoring American immigration laws (regardless of whether or not they are), Your allegation was that Democrats WERE for strict immigration laws.