In the context of the Crusades (which is the only area I'm really comfortable commenting on), the tendency was for Latin Christian authors to refer to the Byzantines as "the Greeks" - because the Empire's dominant language was Greek and for Western writers the Roman Emperor was in Germany - while Muslim/Arabic sources called them the Romans (specifically, the most common term is Rûm - a specialist in Arabic could probably explain the complex history of the term and its other uses, but for our purposes it's a name derived from Roman and in this context also means Roman).

There are a lot of authors writing during this period and there are definitely several outliers who use different terms. I'm not aware of any Latin Christian author using Roman when referring to the Byzantines, but other more circumspect terms do pop up from time to time - things like "our Christian brothers to the East" and stuff like that. The Arabic sources I'm familiar with are pretty much universal in calling the Byzantines the Romans - as distinct from the Western European "Franks" - but my knowledge is limited to a handful of authors who write directly about the Crusades and not the full body of writing produced by Arabic and Muslim authors at this time.

For sources on this kind of thing it's usually best to go back to the original texts where possible. Luckily there's a good number of compiled translations of primary sources that are pretty easy to read. I personally like Chris Tyerman's collection Chronicles of the First Crusade for European authors and Francesco Gabrieli's Arab Historians of the Crusades.