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.
Persian limes are just a cross between key limes and lemons. Before Persian limes were popularized in the 20th century, key limes were just called "limes" in the US (and still are in cocktail recipes and latin markets).