You are correct that http://www.delta.com/planning_reserv...ex.jsp#devices

says:

Assistive Devices

If you need an assistive or non-oxygen-generating life-support device, like a respirator, nebulizer, or ventilator, you can bring it in the cabin for use during the flight, as long as it fits in an FAA approved storage location. You'll need to purchase an extra seat at the lowest available fare if you need it for your equipment.

We can't permit wet-cell batteries in the aircraft cabin for safety reasons. We recommend you use a dry- or gel-cell battery for your equipment, since electrical outlets are not available on our aircraft.

Oxygen concentrators (other than Inogen One and AirSep Lifestyle) and CPAP machines can be carried but not operated during the flight.

Notwithstanding that I'd call Delta and ask about use of a CPAP with a dry or gel-cell battery. I last asked about two years ago and was told yes (flying Business Class ATL-FCO) but decided not to use it. Let us know the answer.

NWA says No to CPAP
 

Last night, on my connecting flight from NRT-SIN, the FA told me that I could not use my CPAP. I have used mine on doens of trips, primarily on UA and SQ with no problem. As is reflected in earlier posts here, airlines have differing policies. It pays to check ahead.