You are correct in noting that historians view the post-1204 Byzantine state as fundamentally different from that which preceded it, and there is some debate as to whether the restoration of the city of Constantinople actually corresponds to a restoration of the Byzantine Empire. Part of the problem, as you mentioned, is the fact that the Empire of Nicaea, the Byzantine rump state that would eventually reconquer the city, was only one competing Greek successor state to the fallen Byzantine Empire. Also rising after the fall of Constantinople to the Frankish/Venetian crusading army was the Despotate of Epirus under the powerful Angelos family. Epirus and Nicaea alternately cooperated and fought, with both polities claiming the role of successor to the Byzantine Empire and trading the important city of Thessalonica, before finally the Epirotes succumbed to the Palaiologan Byzantine state in 1337. This is one factor in diminishing the reconquest of Constantinople as a sign of the Byzantine Empire being officially restored - the fact that it continued to fight against another Greek state also claiming to be the Byzantine Empire. If the Byzantines themselves could claim to be “Roman” without actually controlling the city of Rome, then why can’t the Epirotes claim to be Byzantine without owning Constantinople?

I should also clarify that at no point were any of these states claiming to be “Byzantine.” That term only originated recently as a convenient way to describe the generally Greek-speaking eastern half of the Roman Empire that succeeded the fall of the Western state. In reality the residents of what we call the Byzantine Empire called themselves something closer to “Basileia Rhomanion,” Medieval Greek for “Roman Empire.”

Anyway, back to our story. Further complicating matters for the Nicaean claims to the Byzantine mantle was the fact that another Byzantine rump state, the Empire of Trebizond, survived and claimed lineage from the Roman Empire. However, Trebizond is even more unique in that it actually broke away from the Byzantine Empire even before Constantinople fell in 1204. In April 1204, mere days before the city would be sacked by the crusaders, David and Alexius Komnenos, grandsons of former Emperor Andronikos Komnenos, captured the wealthy Black Sea trading city of Trebizond and proclaimed themselves Emperors. Brought up at the Georgian court after the Komnenos family fled the city following Andronikos’s death, the brothers held a strong claim to the Byzantine throne through their families’ legacy, and while the Trebozondian state never exerted much influence, it can be argued that they maintained the best claim to the Byzantine legacy. The fact that they predate the fall of Constantinople only adds to their claim as a continuation of the Byzantine state.

So, while I hate to give this answer, the best way to address your question is “it depends, and historians continue to debate it.” Since we are already applying the term Byzantine to these states after the fact, there can’t be any real way of classifying them officially. Certainly the Nicaean emperors considered themselves continuations of the previous line of Byzantine rulers, and controlling the city of Constantinople added to that claim, but other Greek rulers in the area felt differently. As for other states, few European rulers ever recognized the Byzantines as their preferred demonym, “Roman.” They mostly called them “The Greek Kingdom,” which only made it harder for the two groups to work together on common ground with such vastly different views on their claims to the glory of Rome.

As to the lack of attention about the Nicaean reconquest of Constantinople, that can at least be partially attributed to the manner in which the city was retaken. By 1261 the fall of Latin Constantinople seemed a foregone conclusion. The city was ruined, depopulated, and dilapidated after a half century of neglect by Frankish rulers. Politically the state was isolated, with the Latin Empire’s only allies, the papacy and Venice, being respectively uninterested and unable to lend assistance. Militarily weak, economically bankrupt, and geographically isolated with Nicaean strongholds to either side. Despite this weakness, turmoil in the Nicaean court led to a delay in any serious effort to take the city. Finally, in 1261 the Nicaean general Alexius Strategopulus, while leading a small army in Thrace to probe at the Latin defenses, learned that much of the Latin garrison of Constantinople had left the city to attack the island of Daphnusia at the mouth of the Bosphorous, in an attempt to open the Bosphorous to allow Venetian galleys to reinforce the city. Acting quickly, Alexius immediately led a detachment of men to a postern gate in the walls, slipping a small number of men through who overwhelmed the few guards and flung open the gates. Neither the Latin Emperor Baldwin II nor the Nicaean Emperor Michael VIII Palaiologos knew the attack was coming, and Michael refused to believe the city had fallen to his forces until he received Baldwin’s personal regalia. He made a triumphant procession into the city, but by that time it was a shadow of its former glory, never again to regain its luster under Greek rule.

For further reading, I highly recommend John Julius Norwich’s “A Short History of the Byzantine Empire” which provides a detailed yet nuanced history of the long history of Byzantium.

In the immediate aftermath of 1204, Nicaea was indeed one of several splinter states as outlined above and there is a fair argument to be made for the Byzantine Ρωμαίοι identity of the population to have contracted to Constantinople itself over the twelfth century.

Provincials had been mistreated by the administration. Taxation was failing because the army was failing, local warlords and dynasties promised better protection in the years 1180-1204 than the imperial army occupied by the Bulgarian uprisings and committed or lost against the Turks at Myriokephelon in 1176. What this triggered was a sense of rejection of Constantinople by many cities and Nicaea really struggled to legitimise itself for the first four years after 1204. The patriarch refused to move to Nicaea, preferring to administer to the dispossessed in Thrace and it was only in 1208 that he either died or resigned his office and allowed for Autoreianos to be installed as patriarch and to crown Theodore Laskaris.

This placed Nicaea at the head of the Orthodox order if not the Byzantine one. Immediately bishops started to arrive from Cyprus and Anatolia wanting recognition of their bishopric and Epiros would spent much time and effort under Demetrios Chomatenos, bishop of Ohrid, trying to prove the legitimacy of their owb autocephalous church. Though they practiced the right to crown an emperor, as Ohrid did when the Komnenoi-Doukai took Thesslaoniki, the act was derided across the Orthodox word as against convention.

The honest truth is that the result was an impasse that was only truly solved when Nicaean strength grew exponentially under John Vatatzes in the 1220s and 1230s and he, in the space of barely two years, doubled the size of the rump. A Bulgarian alliance allowed him to reduce and take Thesslaoniki which was the end of Epirote imperial ambitions. Their church would continue to be at odds with the Nicaean to the extent that Theodore II Laskaris (1254-1258) tried to have the whole Orthodox Church of the Balkans under the Epirots excommunicated.

Several parts of the Byzantine sphere also refused Nicaean hegemony, Rhodes continued under the Caesars of the Gabalas dynasty who have left an infuriatingly small material record, but they were merely continuing the trend already begun by Cyprus in 1183 under Isaakios Komnenos.