Will I need to retake my written qualifying exams?
I don't think this is a matter of finding the right argument.  If the rules of the specific department in question explicitly allow you to keep your old exam results upon readmission to the program after any length of time, then if you are readmitted to the program then you get to keep your old exam results.  If they explicitly say the contrary, then you can't.  

More likely there are no explicit rules which address what happens when a student leaves the program and then is later readmitted to it: there are no such rules in my program, for instance.  In this case it is really up to the discretion of the faculty in the department.  Moreover, since this issue will probably be bound up with your admission decision, being too pushy or argumentative about it could really backfire.  I would say that it is certainly fair to explicitly mention that you passed the written qualifying exams in the same department at such and such a time, that you are excited to pick up where you left off, and that you are wondering whether this will be possible.

I do not advise that you say more than that or that you give detailed arguments.  It is also not a matter of "equitability": they are not obligated to admit you at all, so they cannot be obligated to admit you with certain additional constraints.  To be honest, the harder you argue about this the more strongly I suspect that you should take the exams again.  Nine years is more than long enough to forget everything that you learned as a master's student.  Indeed you write


  While I couldn't pass the present day equivalent versions of these exams (without much study).


Since you have said this, I would seriously consider removing the identifying information from your account!  Your own confidence that you could not pass the current versions of the exams is rather strong evidence that you should take them again.


  neither could many graduate students like me who are further along in their study.


There are almost certainly no other students in the program who passed their exams nine years ago.  Many PhD programs have total time limitations for exactly this reason.  (There was a student in my program who had done everything but his thesis, and he eventually got timed out because he had taken too long: I believe, after nine years.)  Moreover the other students in the program have spent the time since the exam pursuing their studies and going on to fulfill other programmatic requirements.  You left the program.  It is really not an equivalent situation.

Note also that although the mathematical knowledge tested on qualifying exams is not really time-dependent, nine years is long enough for the structure of the exams to have changed: if e.g. the syllabi are different or if the faculty know that the standards have been raised, then it is probably inappropriate to hold a newly admitted student to different requirements than other newly admitted students.  To be fair, it is probably more likely that there has not been significant change.

Finally, let me note that you haven't mentioned why you are not willing to take the exams again.  According to the structure you described, if you retook all the coursework it would take only one more year.  One year in an American PhD program is not a long time at all.  Taking one year to solidify your foundations before proceeding could actually allow you to graduate in less time than if you dive into something headfirst and then only after months or years realize that you need to backtrack to learn the basics.  Please think carefully about whether getting excused from the exams is really in your long term best interest.

Good luck.