He is a former Vala.
As you say, the Maia and the Valar are both just classifications of the Ainur. The full passage you quote is as follows:
The Great among these spirits [the Ainur who came to Arda] the Elves name the Valar, the Powers of Arda, and Men have often called them gods. The Lords of the Valar are seven; and the Valier, the Queens of the Valar, are seven also. These were their names in the Elvish tongue as it was spoken in Valinor, though they have other names in the speech of the Elves in Middle-earth, and their names among Men are manifold. The names of the Lords in due order are: Manwë, Ulmo, Aulë, Oromë, Mandos, Lórien, and Tulkas; and the names of the Queens are: Varda, Yavanna, Nienna, Estë, Vairë, Vána, and Nessa. Melkor is counted no longer among the Valar, and his name is not spoken upon Earth.
Note the tense of the passage, which, in context, is a book of myth and history. No longer is Melkor counted among the Valar, and his name is not spoken on the Earth. This can refer only to the present time (or relatively present times) , but certainly before Feanor 'renamed' him.
Later, at the end of the Valaquenta:
Last of all is set the name of Melkor, He who arises in Might But that name he has forfeited; and the Noldor, who among the Elves suffered most from his malice, will not utter it, and they name him Morgoth, the Dark Enemy of the World. Great might was given to him by Ilúvatar, and he was coeval with Manwë. In the powers and knowledge of all the other Valar he had part, but he turned them to evil purposes, and squandered his strength in violence and tyranny. For he coveted Arda and all that was in it, desiring the kingship of Manwë and dominion over the realms of his peers.
From splendour he fell through arrogance to contempt for all things save himself, a spirit wasteful and pitiless. Understanding he turned to subtlety in perverting to his own will all that he would use, until he became a liar without shame. He began with the desire of Light, but when he could not possess it for himself alone, he descended through fire and wrath into a great burning, down into Darkness. And darkness he used most in his evil works upon Arda, and filled it with fear for all living things.
The Valaquenta says the Elves named the Valar (the membership of the Valar therefore being defined by the Elves), who were simply the 'Great' among the Ainur who entered Arda. So the question becomes a matter of when the Elves would have deemed Melkor to be no longer one of the 'Great.'
We don't know what tales the other Ainur might have told the Eldar of Melkor. If they said anything, I imagine they left it at him being one of the greatest among them, yet still rebellious. Either way, for a time, the Elves regarded Melkor as a great guy.
Then Manwë granted him pardon; but the Valar would not yet suffer him to depart beyond their sight and vigilance, and he was constrained to dwell within the gates of Valmar. But fair-seeming were all the words and deeds of Melkor in that time, and both the Valar and the Eldar had profit from his aid and counsel, if they sought it; and therefore in a while he was given leave to go freely about the land, and it seemed to Manwë that the evil of Melkor was cured. [...] Therefore all the more did he feign love for [the Elves] and seek their friendship, and he offered them the service of his lore and labour in any great deed that they would do. The Vanyar indeed held him in suspicion, for they dwelt in the light of the Trees and were content; and to the Teleri he gave small heed, thinking them of little worth, tools too weak for his designs. But the Noldor took delight in the hidden knowledge that he could reveal to them; and some hearkened to words that it would have been better for mem never to have heard.
(Of Feanor and the Unchaining of Melkor)
This all came to an end pretty quickly.
Then Fëanor rose, and lifting up his hand before Manwë he cursed Melkor, naming him Morgoth, the Black Foe of the World; and by that name only was he known to the Eldar ever after.
(Of the Flight of the Noldor)
So, looking back to the your quote, his name was, at this point, 'no longer spoken on the Earth', at least by the Eldar, who are alleged to have been writing the histories our Hobbits based The Silmarillion on. I think it is logical that at this point, the Eldar would have no longer considered Morgoth one of the Valar.
But you could make the argument that Morgoth was still one of the 'Powers of Arda', Black Foe of the World, or not. In that case, I would say he certainly ceased being numbered among the Valar by the end of the War of Wrath:
The war was successful, and ruin was limited to the small (if beautiful) region of Beleriand. Morgoth was thus actually made captive in physical form,9 and in that form taken as a mere criminal to Aman and delivered to Namo Mandos as judge — and executioner. He was judged, and eventually taken out of the Blessed Realm and executed: that is killed like one of the Incarnates.
Being ignominiously captured, killed, and expelled from Arda would certainly seem to disqualify you as one of the Great Powers of Arda - even if his malice lived on.