How can I tell the difference between strained yogurt and "Greek-style" thickened unstrained yogurt?

Another added thickener can be carageenan, a seaweed.

"Greek Yogurt" is thick because it was strained. As is labne/lebneh, a slightly more sour Middle Eastern kind of thick yogurt.
If a yogurt is "thickened" without straining it, it could be thickened with cornstarch or tapioca or some other non dairy binder. Sometimes goat or sheep's milk yogurts do not have as high a fat content to make them thick and are thickened with different binders.
My yogurt preference is based on flavor. I prefer {unsalted} labne over Greek yogurt because they are interchangeable in my recipes but I find the former brighter and smoother.

Dunno. They should be fairly interchangeable in flavor as well as texture so in theory it shouldn't matter which you have/use.