Just to make things clear (and considering that in my family we make our own oil...small production only for self consumption, so I know a couple of things about it):
When you hear things like "this extra virgin olive oil is better than that because it's the first mechanical extraction of oil (cold pressed)" or similar, it's just marketing...because first mechanical extraction of oil (cold pressed) is by itself the definition of extra-virgin olive oil, so that goes without saying!
I don't know how labelling works around the world, but basically in the EU you are scamming people if you write extra-virgin olive oil on something that is not extracted that way 😀