What type of yogurt is most similar to the yogurt used in authentic Middle-Eastern cuisine?
Greek yogurt is simply strained yogurt. It was only fairly recently that Greek yogurt was widely available in the United States, so prior to wide availability, a simple substitute was to strain normal yogurt. 

The type of sauce you are trying to make would definitely use a strained yogurt. Whether you wanted to strain your own or buy strained yogurt (aka Greek yogurt) is really up to you. In Greece, sheep's milk was traditionally used for yogurt, but now there's a lot more cow's milk too, so either would be authentic. I would think that full fat would be the most authentic, but not the healthiest, as I doubt low fat varieties were used traditionally (if they even existed). 

The American use of the term Greek yogurt is really just one of branding. Many other areas in the middle east use the same style of strained yogurt. The Greeks also have plenty of normal (unstrained) yogurt.