As an Italian (not from Genoa though), pesto to me has always been just a pasta sauce, so I was very surprised to see it used on salads and pizzas abroad. But recently (in the last 2-3 years) I’ve seen quite a few Italian pizzerias (not the most traditional, of course) starting to use it as an alternative choice to the tomato sauce, or, sometimes, in place of the olive oil on top. Not sure if it’s a regional variation gone national or an influence from abroad... but if it’s the latter, no Italian would ever admit that :)
This is either lahmadjun or pide. They are close, but don't seem to have common origins, just independent discoveries. Same as flatbreads. And putting some topping on one is hardly a world shattering idea.
I find them in Turkish restaurants around here as well. They often look like this: http://loveantalya.com/turkish-pide-with-meat-filling-cheese...
Pizza-like but not pizza. I love them for what they are. But again I love any kind of pizza (Naples, Chicago, thin, thick, doughy etc.)