DMAIC is a far superior process improvement framework / methodology than PDCA, right ?

With the utmost respect to my fellow professionals I cannot for the life of me understand why they still want to promote PDCA when in fact DMAIC is a much more well balanced, logical, easier to teach, easier to follow methodology.  For a start the word ‘Plan’ is too generic a word.  Are we not ‘Planning’ and ‘Refining the plan’ all the way through a project ?  You will recognize that ‘Plan’ is an activity the is only one activity recognised the PMBOK of the Project Management Institute,  started in the ‘Initiate’ phase.  The project plan is constantly being updated.   In fact ‘Define’ is a what we are trying to do as a first step in any project.  Define the problem we are trying to solve.  Define the stakeholders. Define the project team.  Define the scope.  Define a high level plan.
The next biggest issue I have with PDCA is that the word ‘Do’ and the word ‘Act’ are so similar in meaning.  Why do we have One Quarter of the Methodology ( 1 letter out of 4 ) describing the activity of moving onto the next thing ?  Would not the word ‘Close’ be better ?  Do we really need to take up 25% of the letters for this step ?  In DMAIC the word ‘Control’ is a far better description of the post-Implementation step.    I believe the Walter Shewhart and W. Edwards Deming would both agree that DMAIC is a better description of what we really need to be doing in a Process Improvement project.
Can you really defend PDCA when it goes up against DMAIC ?   I would be interested to hear how you can do that ?