I would say that Martin was an exact name of MLK but not the exact name, and I would say that neither "Martin Luther" nor "Luther King" was even an exact name because treating "Martin Luther" as a name means taking "Luther" as a surname and treating "Luther King" as a name means taking "Luther" as a given name, and neither is correct for MLK.

Of course, you may choose to use these terms differently. Perhaps you couldn't truthfully say "neither of those is his actual exact real name". But I could, and I did.

(I would also have preferred not to get into this particular dispute because although it's not exactly hard to find out Scott's actual exact real full complete official name -- or at least something that I take to be that -- I would have preferred to offer a little less assistance to anyone trying to do it. But no matter.)