Why did the term "Byzantine Empire" enter common usage instead of "Eastern Roman Empire" or "Roman Empire"

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).

Warren Treadgold, one of the most eminent scholars on the Byzantine Empire puts it simply as follows:

Modern historians have called this empire "Byzantine" because it was ruled not from Rome but from Constantinople, the former Byzantium

Hieronymus Wolf was the earliest known historian to use the name of the Byzantine Empire's capital to refer to the empire itself. As far as we know, there where no other "ulterior motives" other than to simply distinguish between an empire in two different stages of history. In the eyes of historians, the Empire was always named after its capital city. So since the Roman Empire was without Rome for a great part of its history, the name of its later capital was used to replace it.

Other terms for the Byzantine Empire, particularly "The Greek Empire" or "Empire of the Greeks" (Imperium Graecorum) was used by 19th century historians interchangeably with "Byzantine Empire" where appropriate. Mostly when highlighting the ethnic or linguistic differences between the Byzantine Empire and its western neighbours. One notable historian who used "The Greek Empire" was George Finlay.

As for Gibbon, like other classicists he was not too fond of the Byzantine Empire. He saw it as a betrayal of everything that ancient Rome and Greece stood for. (see Norwich). Although still accepting Byzantium as a continuation of the Roman empire, he is well known for trying to distance his ideal Roman Empire from that of "barbaric" Byzantium by highlighting key differences. Differences which were the cause of the Empires downfall. He did not however promote "Byzantium" as an alternative to "Roman" as such practices by his time where already widespread in western Europe.

In summary, the use of the term "Byzantine" came naturally and spread through Europe over time as a way for historians to conveniently distinguish the later history of the Roman Empire to its early history. The negative connotations associated with the term came much later.

This book goes significant detail of the reception of Byzantium in western europe after its fall. I have not read it but it might be worth a shot.