I’m leaving a note to explain my downvote here: Really? The NY Times is spamming a home sale? Out of the entire article, the home sale is mentioned once, at the end, as a journalistic detail. Don’t be like that.

No, it will not happen. The only way you could possibly do this would be to purchase another home while you still own this one. Given that you can barely afford the current payments, you would not qualify for another mortgage at this time. Once you go through a 'short sale', you will not qualify for another mortgage either, since a short sale will damage your credit rating just as badly as will a foreclosure. If you elect the short sale avenue, expect to wait three to four years before you qualify for another mortgage.

Bank owned is a foreclosure.
The original owner has lost his home to the bank. Bank Owned

Short sale means that the owner still has possession of the house.
He has "struck a deal" with the bank to walk away from the home at the price the home sells. This is done by owners that are facing foreclosure, but not yet there.

Keep in mind that with either one is is very hard to negotiate down a price.
Banks are not very willing to go down in price as regular sellers that may be desperate