Or did they use a qualifier, like East Romans? Did they consider them as separated countrymen? Or had the earlier separation of the Empire created a distinct feeling of "us and them" towards the Byzantines even from the outset of the fall of the West? When did even this tacit linking of a shared identity end?
It would very much depend on the era. From the 530s into the 8th century, Rome itself was Byzantine territory (sometimes more and sometimes less autonomous), and, in fact, the last Roman Emperor to build a monument in the city of Rome itself was Phocas (r. 602-610), a hundred and thirty years after the nominal end of the Western Roman Empire.
As Lombard power waxed and Byzantine power waned in Italy in the 8th century, the Papacy increasingly had to look to itself for its own defenses, and by the mid-8th century started turning to the Frankish kings Peppin the Short and his son Charles (Charlemagne) for protection from the Lombards rather than the Byzantines. The Iconoclast crisis in the Byzantine Empire gave Rome even more leeway: the moral authority of the Papacy (even though few took its claims to be the supreme leader of the Church, everyone took its claim to be one of the supreme leaders of the Church seriously) in condemning first the Iconoclasts, and then Irene's Iconophiles for going too far in the other direction, gave the Papacy real ideological separation from the Byzantine Empire.
The Papacy's recognition of Charlemagne as a revived Western Roman Emperor on Christmas Day, 800 was not only a reward to Charlemagne for protecting the Papacy in general and Leo III in particular, but also a calculated slap in the face to the Byzantines. Even during the Byzantines' brief revival of their Italian ambitions in later times, Rome itself looked West rather than East.
The irrevocable split between the Church didn't occur until the 11th century, and even afterwards, the understanding remained (at least in the West) that the Roman Catholic and Eastern Orthodox Churches were one church in a temporary schism (that has just never ended) over control of the Church and several minor doctrinal issues (the filioque clause is a famous example) rather than two permanently different entities.
Part of the problem with your question is it covers an impossibly long timeframe (a millennium). The attitude of Rome and the rest of the West to the Byzantines at a time when they controlled Rome and most of Italy (the 6th and parts of 7th centuries) vs. the time when the Byzantine Emperors were seen in the West as pathetic beggars pleading for a lease on life for their Empire (14th-15th centuries) is huge, and the difference in prestige of the Eastern Empire is obvious between those two extremes.