I've always just used the border where Russia starts as Asia, and Turkey's border with Syria and Iraq as the start of the middle east.
I used your system, until I realised Russia is considered in Europe. They play in the European leagues not the Asian leagues (sports). As for Turkey, I thought the same too but when I rethought about it I guess Turkey is in Europe. I believe it used to be headquarters of the Roman empire, and the Romans were not middle-eastern.
Russia being considered part of Europe started with Peter the Great in the 17th and 18th centuries. He went and studied under a European military, I think the Prussian. After that he modeled almost everything, from politics to fashion around the European style of the day, not to mention they more physically resemble Europeans than Asians.
Turkey is another problem all together. Byzantium/Constantinople/Istanbul was set up as a second Rome because of the strategic placement. Nearly every trade route went through there. When Rome fell, the empire continued on through Byzantium, even though they spoke Greek. They have faced a history similar to Israel and Palestine because of the foreign influence/rulers and native contempt.
It's an old fashioned term, but Turkey, Israel, Syria, Jordan and all that lot sometimes get referred to as the near east.
I'm reasonably sure the reason Turkey and Israel play sports, specifically football, with the european countries is to avoid potentially explosive matches with Iran, Iraq and the rest. I suspect TV revenue plays its part as well.
Sir Gawain summed up Russia well. From what I remember from my history classes, at one point the Russian aristocracy considered their own mother tongue vulgar and instead spoke exclusively in French. The revolution saw Russian move back towards seeing itself as more slavic. It's such an huge, unique and ancient country that, like Turkey, doesn't fit neatly into any category.
Turkey can play in Europe maybe because some part of it is in Europe, Israel plays in Europe I believe because all the Arab countries threatened to quit if Israel was to play with them in Asia. At least, thats what I heard what happened in the Fifa.