There were a series of names used during the Middle Ages for the Roman Empire in Constantinople by Western Europeans. Like you said the popularity and use of these terms would change over time. often depending on the political circumstances.

There's not much evidence of the Empire not being acknoweledged as Roman prior to the 8th Century as far as I am aware. This was during the time of the Byzantine Papacy, in which Rome, and as a result the Pope was under formal Imperial control. As such the Popes themselves, as part of the Empire did not have any political or religious reasons for distancing themselves and the Empire from its Roman past.

However with the advent of Iconoclasm in the 8th Century, relations between the Papacy and Constantinople became much more tense. The Pope would not accept what he saw as the heretical, brutal act of destroying icons and effectively severed ties with Constantinople as a result.

Looking for new allies, the Papacy looked north, towards the emerging Kingdom of the Franks, who became the new protectors of Papal independence. Upon the coronation of Charlemagne, the Franks were seen as the one, true Empire, which meant that Constantinople simply no longer could be accepted as the one, true Roman Empire.

From this point gradually the term "Empire of the Greeks" or "Kingdom of the Greeks" (Latin: Imperium Graecorum) becomes more common among the Western Europeans. Liutprand of Cremona during his visit to Constantinople would stubbornly refer to the Emperor as the "Emperor of the Greeks", which caused the current Emperor, Nikephoros Phokas to apperantly become extremely irritated. Liutprand records the following exchange:

But, to increase my calamities, on the day of the Assumption of the Virgin Mary the holy mother of God envoys of the apostolic and universal pope John, through whom he asked Nicephorus, the emperor of the Greeks, "To close an alliance and firm friendship with his beloved and spiritual son Otto 'August Emperor of the Romans'." Before the question as to why--this word, this manner of address, sinful and bold in the eyes of the Greeks, did not cost its bearer his life--why he was not annihilated before it was read, I, who, in other respects, have often shown myself enough of a preacher and with words enough at my command, seem dumb as a fish!

The Greeks inveighed against the sea, cursed the waves, and wondered exceedingly how they had been able to transport such an iniquity and why the yawning deep had not swallowed up the ship. "Was it not unpardonable," they said, "to have called the universal emperor of the Romans, the august, great, only Nicephorus: 'of the Greeks'. A barbarian, a pauper: 'of the Romans'? Oh sky! Oh earth! Oh sea!"

"But what," they said, "shall we do to those scoundrels, those criminals? They are paupers, and if we kill them we pollute our hands with vile blood; they are ragged, they are slaves, they are peasants; if we beat them we disgrace not them, but ourselves; for they are not worthy of the gilded Roman flail and of such punishments. Oh would that one were a bishop, another a margrave! For sewing them in sacks, after stinging blows with whips, after plucking out their beards or their hair, they would be thrown into the sea. But these," they said, "may continue to live; and, until the holy emperor of the Romans, Nicephorus, learns of this atrocity, they may languish in narrow confinement."

This shows that Western Europeans evidently took the term "Empire of the Greeks" and Constantinople's lower status compared to the Holy Roman Empire very seriously, and likewise that the Emperors in Constantinople looked at this outlook with great disgust.

Other terms were also used occasionally such as "Romania", which was a term that was also used by the Byzantines themselves informally. When the Latin Crusaders captured Constantinople in 1204, the treaty that divided the Empire into new Crusader States was named the "Partitio terrarum imperii Romaniae" (Partition of the lands [belonging to] the Empire of Romania). The subsequent Latin Empire that succeeded Roman rule was also contemporaneously as the Empire of Romania (Latin: Imperium Romaniae).

The term Romania was thus especially used during the first half of the 13th Century, when Constantinople was in Catholic hands, as the Crusaders wished to establish a legitimate, Catholic, Roman Empire of Constantinople for themselves.

Meanwhile the Holy Roman Empire was at the time known to Western Europeans simply as "Imperium Romanum" or the Roman Empire. The "Sacrum" (Holy) suffix wasn't added until 12th Century by Emperor Frederick Barbarossa in order to legitimize his ambitions to dominate Italy and the Papacy, and to put his secular power over his realm above the Pope's spiritual power.

I hope that answered the question pretty clearly, if you have any further follow-up questions, I'll gladly attempt to answer them. If you feel something I've stated is unclear then I'll attempt to rephrase it more clearly.