But these articles have all glossed over another important fact: defunding Obamacare would defund Obamacare.
Guess what doesn't require annual appropriations? Almost all of Obamacare. So, yes, B-Berg, NatJo, and WaPo, a government shutdown won't defund Obamacare.
But that's not the whole story. The House passed a CR numbered H.J. Res 59. This House CR, in addition to funding the government at current levels, also contains a defund-Obamacare provision. This provision at least aims to defund Obamacare, and I think it does:
Notwithstanding any other provision of law, no Federal funds shall be made available to carry out any provisions of the Patient Protection and Affordable Care Act...No entitlement to benefits under any provision of the Patient Protection and Affordable Care Act... shall remain in effect on and after the date of the enactment of this joint resolution, nor shall any payment be awarded, owed, or made to any State, District, or territory under any such provision.