In the 1800s, apparently, historians regarded the Frankish Empire to be an extension of the Holy Roman Empire back toward the original Western Roman Empire. Since then, the idea of the Frankish Empire being a continuation of the Holy Roman Empire back in time has not been held in favor (which makes sense in that the Carolingian Empire was in fact a forerunner not only to the HRE - itself a precursor to the Austro-Hungarian Empire -, but also to France, which began evolving toward separate entities with the Treaty of Verdun in 843).
Indeed, the name Holy Roman Empire was not in use until the 12th century (Frederick I "Barbarossa" was the first to use this title while still alive). However, it is accepted today that the entity that became the HRE - basically an empire of the German states - began with the crowning of Otto I.
The continued use of HRE in sources today to depict the Frankish Empire before 962 is primarily from the use of such 19th century-based sources as Burke's Peerage, which is still very useful for English nobility, but not so accurate for as far back as the Frankish Empire. Continued use today is an indicator of a lack of understanding of the limitations of that source (Darryl Lundy's Peerage website makes particularly notable use of that source throughout the site, and so all of Lundy's work related to the Frankish Empire is likewise inaccurate - as the phrase goes, "garbage in, garbage out").
Of course another secondary source that is notorious for being a classic example of GIGO is Wikipedia, but still, the English Wikipedia page for the Holy Roman Empire seems to be well-backed by other sources. The second paragraph provides as a source for the HRE beginning in 962 as an Italian Encyclopedia, which itself contains a number of secondary and a couple primary sources cited in the section depicting the life of Otto I, explaining why he is considered to be the first Holy Roman Emperor:
The other nit worth picking here is also the significance of the year 1512. Before this, the state was translated in English as the Holy Roman Empire. After the Diet of Cologne that year, the name changed to the Holy Roman Empire of the German Nation (a person can find the appropriate translations in German and Latin on Wikipedia). This state lasted (sometimes nicknamed as "Austria") until its surrender to Napoleon in 1806 with the Treaty of Pressburg.
In short, Holy Roman Emperor doesn't apply to the pre-Otto Emperors. Charlemagne himself accepted the title of Emperor of the Romans, as apparently did Louis the Pious. Other "Emperors" could be considered successors to this title, though the title was probably not in use (please correct me if I'm wrong on that one). Though Frederick Barbarossa might have looked to Charlemagne for his legitimacy as the first self-proclaimed Holy Roman Emperor, this title can only be posthumously extended back to Otto - no further than that. Earlier than that, the Franks lived within a larger Frankish Empire, a different entity, one that included France as a component.