A Key Lime is only vaguely a lime (they call it a Dayap and I don't think they would consider it a lime at all), that's why we call them 'Key Lime' instead of just lime. I'm putting a fine point on it, but I think when we say 'Lime' we mean something like a Persian Lime. You don't make Persian Lime pie!

As far as I know in the Philippines they don't have a common word to describe a Persian lime, and I never saw one in a store there.