The historiographical description of the use of the term is presented in Wikipedia:
The first use of the term "Byzantine" to label the later years of the
Roman Empire was in 1557, 104 years after the empire's collapse, when
the German historian Hieronymus Wolf published his work Corpus
Historiæ Byzantinæ, a collection of historical sources. The term comes
from "Byzantium", the name of the city to which Constantine moved his
capital, leaving Rome, and rebuilt under the new name of
Constantinople. The older name of the city would rarely be used from
this point onward except in historical or poetic contexts. The
publication in 1648 of the Byzantine du Louvre (Corpus Scriptorum
Historiae Byzantinae), and in 1680 of Du Cange's Historia Byzantina
further popularised the use of "Byzantine" among French authors, such
as Montesquieu. However, it was not until the mid-19th century that
the term came into general use in the Western world.
I have formulated the problem of the negative connotations of the term “byzantine”, as a separate question. But, as the OP mentioned “ulterior” motives behind the use of the term (suggesting some anti-Byzantine bias — and because the other answer dismisses that suggestion rather hastily, in my opinion) a distinction needs to be made between the choice of a term like “the Byzantines” to denominate Constantinople (no matter the ulterior motives or the bias behind that choice) and the negative connotations that the term “byzantine” as such could have acquired: the fact that (because of some anti-Byzantine opinion) the attribute “byzantine” may have acquired in time some derogatory meaning is a different problem from that whether the term in itself was originally adopted with some negative connotation/intention.
Also, there is a difference between having a bad opinion about "the Byzantines" and using the term "byzantine" as a derogatory one. (I don't yet know about each of the authors that disliked the Byzantines whether they also used the term "byzantine" in that sense.)
Considering Gibbon, he was not alone and not the first to promote a negative opinion of “the Byzantines”; in that he was following the general trend within the specific ideology of the Enlightenment — which also coined the meaning of “Eastern Europe” (with heavy influence to this day) (cf. Inventing Eastern Europe: The Map of Civilization on the Mind of the Enlightenment by Larry Wolff).
By attacking “the Byzantines” Montesquieu and Voltaire were indirectly attacking the French absolutist monarchy (or "the Byzantines" were collateral damage during their war against autocracy). A French monarchic connection with Constantinople had been indeed promoted positively under
Louis XIV (when the collection of historical sources La Byzantine du Louvre was created), and already during Francis I:
France is, in a way, the country of the birth of Byzantinology under
Francis I, although it did not initially welcome many Byzantine
scholars: a Greek collection was created at the Library Royale de
Fontainebleau, by purchasing manuscripts in Italy, copying Bessarion
manuscripts or manuscripts preserved in Rome. This birth was marked by
a strong ideological component: in 1547, the royal presses published
the eulogy of Dionysius the Areopagite by Michael the Syncellus (761 /
762-846), based on a 10th century manuscript. The aim was to
strengthen the legitimacy of the Dionysian story on which the Abbey of
Saint-Denis is based, the burial place of the kings of France…
(Pourquoi Byzance ? Un empire de onze siècles, by Michel Kaplan, p.40-41)
This means that roughly at the same time as the first mentioning of the term, the Byzantine sources were seen in a positive light, as possible ways of legitimizing the divine right of the French monarchs.
On the other hand, the initial choice of “Byzantium” instead of Constantinople is odd. It doesn’t seem neutral either. The initial questions are not yet answered.
Why not Constantinople? Why not “Greek empire”? I have found an explanation presented by the Greek Byzantologist Helene Ahrweiler at the France Culture radio. (The discussion is in French, accessible on the France culture website and on Youtube).
She states that, when “the Byzantine” culture entered the field of interest of the 16th and 17th century humanists, it had to be placed on an intellectual map defined by the exalted ideal of the ancient Greece and the ancient Rome. That created the problem of differentiating “the Byzantine” from “the Greek” (meaning classical Greece) and from “the Romans”. This logical necessity for differentiation was doubled by an axiologic necessity imposed by the negative judgment that Renaissance humanists (and their classicist and rationalist successors) made on the Medieval era, which “the Byzantine” seemed to represent in the East with similar connotations that the term “Gothic” had for the West.
(I haven’t read her books, nor have I found this interesting opinion presented in the books I did read, so I hope to add more on this in the future.)
She also says that old Catholic anti-Byzantine bias is still reflected in the historiography of Catholic tradition by the use of the term "Byzantine", while those of Protestant tradition prefer "Late Roman Empire" (the "Anglo-Saxon") or "Oriental Roman Empire" (the Germans) That might be true in the present, otherwise the English of 18th and 19th century were as ill-disposed as anybody (as quoted here).
No matter the sense of the original choice of the term, its fate varied: in the 16th and 17th century France we see that it could be used with neutral and positive connotation (and even reverently). "It was ultimately the Enlightenment that earned Byzantium its poor reputation" (M.Kaplan, p.44).