The House did go first, The senate then did their thing. Then, just before reconciliation, Scott Brown was elected. So the House just accepted the Senate bill.
The Dems would be claiming that the bill ultimately passed (or at least the main part of it) originated in the House, but really what originated in the House bore no resemblance whatsoever to the final product. In their thinking, the original House bill was not the massive House health care bill that passed just before Christmas.