Putin has a vision of the world he wants its with a strong Russian influence. He has allies who will help him to get that. He also has enacted a road map to a restored Russian empire.
That's one interpretation, but I don't think it's based in Realpolitik.
Both Russia & China have stated their desire for a Multipolar world. What we have now is US Hegemony. Some like to call it Imperialism but that's stretching it. And I note all of this without a judgement one way or the other.
All of these conflicts save perhaps GW1 & Afghanistan, which was a legitimate Counter Attack by the US for 9/11, are pulley wars to maintain or sway influence in a given region. If you examine these conflicts without national bias, you will see many have arisen from Overt or Covert US action. In the most direct case involving the US-Russia balance of power pre-Syria I would reference Georgia.
Where Putin touches it in his Syria Op-Ed and I agree, is the danger of unrestricted use of force outside UN Resolution or Direct Self Defence of ones Nation and/or citizens.
If major nations in the security council (US, Russia, China or other) decide to forgo UN International Law to pursue their agenda's then it's Pandora's Box waiting to be opened. Because now it's not just the US and Russia vieing for power as in the Cold War, but add China, India, Japan and many others. Other nations will take from the example of operating outside the UN and we'll have the League of Nations and chaos all over again.
Cheers,
Zool