Thank you Harry! This clarify everything a little to me. Apparently the Key lime is the less widespread one, but has an unique flavour. I have never seen the fruit being sold here. The Persian lime, instead, apparently is the one i usually find as "lime" in grocery stores. The other ones are sweet limes, seldom used for recipes.
Il try to get a plant with both varieties grafted, so i'll get fresh limes for my recipes!
There is also a Red Lime (don't have the scientific name, will try and get it for you), excellent but I don't think you would have availability to it.
I would not say the Key Lime is less widespread, "originated in southern Asia and was carried by the Arabs across North Africa into Spain and Portugal. It was brought to the Americas by Spanish and Portuguese explorers in the early part of the sixteenth century where it escaped cultivation and became naturalized in parts of the West Indies, some Caribbean countries, and southern Florida."
I would not look for a cocktail tree with both the Key, also known as Mexican or West Indian lime, and Persian lime, also known as Tahiti lime. They have two totally different growth habits with the Persian being heartier and the Persian will most likely dominate in the long run.
