It's not a bad starting point, but it was a bit runny and had too much cumin for my taste.
The ayran that I became accustomed to in the Adana area of Turkey seemed to be about half water, half yogurt and slightly salty. (This is a guess. I'm looking up ayran recipes because I don't know the exact ratios. I didn't find what I was looking for with this recipe.) I don't know if they used lemon juice, but I never saw cumin in it.
This recipe makes an interesting variation of it, but I'd use half the cumin and water. Keep the salt and lemon juice.
Naturally, my opinion is no more valid than any other, but I do think that the cumin will prove too strong for most readers.
We loved all slight variations of Ayran that we tried in Turkey, but this is much different than any of those. I used less cumin (~1 tsp) and it was still a really strong flavor. A more authentic version seems to be simply equal amounts water and yogurt with a bit of lemon and salt to taste.
I have no idea if it tastes like the real thing but it does taste good. The cumin is not overly intrusive and the lemon gives a nice refreshing citrus touch.