Yes, your correction is welcome. They represent their district, the people of their district... and then their state. These days the districts are very intertwined economically and socially, so while each one should be primarily focused on the people of their district, they can and should use their collective influence for the betterment of the state at large.
As an example, if California chose to disassociate itself from the two parties and elected only members of "The California Party" it would command 12% of the votes of the House. That's a lot of leverage wasted because they are divided by party.