Definition of Done (DoD) is for user stories
In a scrum project it is desirable to have a DoD with different levels
No. In Scrum, Definition of Done (DoD) is for user stories only and it remains same for all user stories. When all stories of the sprint backlog cover the DoD then you have completed your sprint.
OK to have a 'less than ideal' DoD
An ideal DoD would contain everything that the team needs to perform in order to build a potentially shippable product every sprint. However based on the context, situation, and team's skills a subset can be chosen as initial DoD. Simplify your DoD to items which you can ensure for all user stories and stick to it. This would mean that at some point you'll have to work on the left out portion of the ideal DoD (this is mentioned as Undone Work in the article mentioned by OP). With the passage of time, you may add more items to your DoD list making it more closer to the ideal DoD.
Tracking Undone Work
Some teams make use of a Hardening Sprint or Release Sprint. Schedule a hardening sprint before doing a release, here you can have your Undone Work (tasks left out from the ideal DoD) which can now be tracked as a regular sprint backlog.
Make sure that a release sprint is not used to bucket all the remaining bugs and other stuff which should have been handled during regular sprints. In the words of Mike Cohn:
The release sprint is not a dumping ground for sloppy work; rather it is a place where some hardening of the system can occur.
Frequency of release sprints
As per Scaled Agile Framework:
Agile Teams can place hardening sprints anywhere, if and when, they are needed. So long as you are not “doing the stuff you should have done earlier,” and you are continually lightening the stuff in the “should be doing earlier” pile, then you are increasingly agile and on the right track.
Note: your idea of maintaining different DoD levels has been mentioned here: https://www.mountaingoatsoftware.com/blog/multiple-levels-of-done