Okay Folks, here's what ASME VIII says...
MAWP = pressure specified by the purchaser for the design of the vessel. It is the pressure at the top of the vessel in its operating position. The MAWP is what is stamped on the ASME nameplate. Additional comment at the bottom of response.
Design pressure (P)= the MAWP plus static head pressure due to liquid (if any) that is used to design each PART of a vessel.
To the best of my knowledge MAOP is not an ASME VIII Code-defined term.
There is an alternative to the value of MAWP stamped on the nameplate. Instead of using the MAWP specified by the purchaser, the MAWP of the vessel may be reported as the highest MAWP permissible, based on the the as-built thickness of each component. This requires that the vessel be designed and plate thicknesses selected. Then the calculations are "reversed" using the a-built thickness (minus CA) and determining what MAWP would be permitted for each part. The MAWP for the vessel will then be the LOWEST MAWP of each part, reduced by the appropriate static head presseure for each part.
Wow, that was quite long winded.
Joe Tank