As you see there's no clear agreement on the one and only way to go. Every bank,with some incentive from the Obama plan if they'd accepted TARP, and others voluntarily, are trying harder to work with homeowners to help them stay in their home if at all possible.
As it is the exact same paperwork, I suggest you exhaust all options to request a loan modification, or refinance first. If you choose to place it on the market as a short sale subsequent to hearing from the bank that the first two options cannot work for you, that would work. You could also try to short sale it simultaneously. HOWEVER, until you have the banks blessing, you become one more of the short sale listings that are really not approved by the lender so, in my mind, not a legitimate listing. These are the short sales that are muddying up the market.
My experience with my short sale clients has been very quick in doing the first approach. In each situation, all with two different lenders on the property, we received approvals within TWO WEEKS of receiving an offer, not some long drawn out 2-3 months. And my sellers understood that we had evaluated all alternatives thoroughly prior to proceeding to short sale.
Whatever solution works, submitting a clear concise package to the lender will get you a quicker resolution.