I always got confused with the name-things. I mean, Melkor (Belegűr, Mailiko, Meleko, Mánfréa, Orgel, Maeleg, Moeleg, Alkar, Melekô, Melkóre, Belegar, ... in earlier forms) is definitely an elvish word, coming from the Primitive Elvish word "MIL-IK-" ("desire, greed, lust"

- so it must have been given to him by the early elves, like all names of the Valar and of course also Ilúvatar himself. Tolkien also states that the Valar just took names (and shapes) to look less alien and foreign to the Eldar, so this must have been added to the Ainulindale in the version of the elves. I don't think that Ilúvatar calling Melkor "Melkor" (and Ulmo Ulmo and so on) is really what "happened", its just the truth for all living beings in middleearth as there is no better one (confusing sentence... but based on the sentence "There are always four sides to each story: your side, their side, the truth and what really happened." (Rousseau))
So Melkor is his real original name in the thought of the Eldar though this is most probably in fact not the case.