When exactly does the date start to qualify for the Overseas Service Ribbon?

Better question: who gives a shit?

It definitely starts the day you swipe your ID card in country. Your unit should track, and begin processing the award as soon as eligible. You unit S-1 should be able to print you a copy of your deployment, or overseas, validation time. I can't remember what it is called, but it literally shows your valid dates of deployments, or proof, of time served. More than likely, if your award isn't processed while in country, then it will be populated off of your deployment portion of your ERB.

According to HRC as of 22DEC2016..

"Note: Overseas tour completion is determined by AR 614-30. If a Soldier is credited with a normal overseas tour completion, then the Soldier may be awarded the OSR. Tours lengths for Iraq and Afghanistan have NOT been established. All requests for exceptions to policy will be processed through the proponent of AR 614-30."

https://www.hrc.army.mil/content/Overseas%20Service%20Ribbon%20OSR

And to the best that I could interpret in AR 614-30, Overseas Service, Table 3-2 and Table 3-3, the tour starts when you leave CONUS and requires 45-60 days of consecutive service in the AO depending on the circumstances.

Most of it seems heavily dependent on where you go and the duty status that you're in. I've been awarded 2 and they were for tours of 9 mos and 4 mos in country. The 4 mo tour was supposed to be 9 mos but a letter came down from higher saying specific units and assets were to vacate the theater by such and such a day.