Ah, the new $600 thingy is an Additional Recovery Rebate.  It is codified separately under §6428A and is to be added to the Recovery Rebate provided for under §6428.

The House bill, coined the "CASH Act of 2020", to increase the quantum from $600 to $2,000 is simply an amendment to §6428A, which means the $2,000 will just be the updated amount for Additional Recovery Rebate if the bill is passed by the Senate and signed into law by Trump.