Doogh/Ayran from scratch
I have no idea why it starts with Greek yogurt. Wherever I've had it - at home, restaurant-made, or ready-bought, it contained just plain yogurt, water and salt. (The ratio varies to taste). This includes ayran made in traditional Turkish restaurants run by Turkish owners. Also, I can't think of a practical reason why true (strained) Greek yogurt could make a difference. The whey doesn't have a strong taste on its own, and what little it has, it is similar to the yogurt itself, so you probably won't notice a difference. You are just losing the nutrients from the whey. 

In some countries, true Greek yogurt isn't available, and what you get sold as "Greek yogurt" is actually sour cream with double the fat content, but with the same protein and lactose content as plain yogurt. This could have some taste difference in comparison to ayran made with plain yogurt, but in my eyes, it will be a negative change. If you use the same yogurt:water ratio as with plain yogurt, you will get too much fat, which isn't pleasant in ayran (which is mostly a refreshing drink). If you use a doubled water ratio so the fat amount stays the same, you get much less taste. 

Conclusion, go with the normal yogurt. It is not only cheaper, it is also the traditional, time-proven way of getting tasty ayran.