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.
To sweeten it, add your favorite sweetener: sugar, honey, agave... whatever you prefer. This is important, because sweetness will help bring out the flavor of the fruit.
General advice:
If you have a fruit you want in your yogurt, pick the form (chunks, pureed, mashed, juice) that you want and mix it in. The Chobani yogurts appear have a variety of these forms, always along with some sugar. If you want banana flavor, sure, mash up a banana and mix it in. Or if you want chunks of fruit, chop it up. Assuming the fruit is soft enough, for this purpose, mashing with a fork is probably a good substitute for pureeing; you'll get some small chunks and some juice.
In response to the updates:
if you want actually fruit-flavored yogurt, then you need juice and/or puree. If you don't have any other tools, mash it as best as you can. If you have a small food processor or blender that will work with small enough quantities (or are making big batches), blend things up and stir them in. Juice and puree will disperse quite well through the yogurt.
Bananas mash easily, especially ripe ones, so just do that with a fork. Strawberries aren't as soft, but you should get enough juice to flavor with if you mash them. Pineapple is tougher. If it's juicy enough, you can probably get some juice out by mashing it; otherwise you might be better off with a food processor or blender. (If it's from a can, you can use the liquid from the can, too - that might also have sugar in it.) Depending on how thorough you are, you may want to use the juice and discard the fibrous parts that are left over.
Best Answer
Greek yogurt has more fat than a "normal" yogurt, about 10%.
Further an original Greek yogurt is made from sheep's milk since there aren't many cows in Greece. This might taste a bit odd for people used to cow's milk though...
When buying Greek yogurt made from cow's milk I recommend you look out for the native brand ΦΑΓΕ.