How would you define extra virgin olive oil? If the term "first cold pressed" comes to mind, then you wouldn't be alone -- after all, the term is splashed across most bottles categorized as extra virgin.
But here's the thing: the vast majority of modern, high-quality extra virgin is not actually pressed; the olive oil is instead extracted from the olive paste via a centrifuge. While olive oil used to only be pressed, pressing exposes the oil to more air, thus accelerating its oxidation. Modern methods minimize oxygen contact, which means the resulting extra virgin is healthier, more vibrant, and shelf stable for a longer amount of time. (A very, very, very small quantity of producers may still press their oils, but generally this is for the sake of tradition.)
Also, when it comes to extra virgin, there isn't a second or third run of anything -- it's just one run and you're done. Extra virgin by default is the juice of freshly milled olives.
Finally, "cold" is a relative term for the temperature at which olive oil is produced. All extra virgin should be produced at around room temperature, so you could say it’s “cold” compared to other types of oils that have been heated up and refined.
To make a long story not so long: when you see the term "first cold pressed," know that it does not define how most high-quality, modern olive oil is produced. The term is typically used because it's what consumers have come to expect, and because the industry is accustomed to using it.
So what term do we use? We "mill" olive oil -- because olives are milled at the mill, in a process that is overseen by a miller.