An example of the date being 962:
My high school AP World textbook, The Earth and Its Peoples Holy Roman Empire: loose federation of mostly German states and principalities, headed by an emperor elected by the princes. It lasted from 962 to 1806. It also says: “In 962 the pope crowned the first ‘Holy Roman Emperor’ (Charlemagne never held this full title).” Charlemagne: King of the Franks (r. 768-814). Through a series of military conquests he established the Carolingian Empire, which encompassed all of Gaul and parts of Germany and Italy.
An example of the traditional 800 date:
A Little History of the World “On Christmas Eve, in the year 800, when Charlemagne was kneeling in prayer in the great church of St Peter’s in Rome, the pope suddenly stepped forward and placed a crown upon his head. Then the pope and all the people fell on their knees before him and proclaimed Charlemagne the new Roman emperor, […]. But now [Charlemagne] wore the crown and was the first German emperor of the Holy Roman Empire, as it later was known.” (You probably detect bias here. The author also wrote, “Yet in all his campaigns [Charlemagne] never lost sight of his goal: to bring all these various Germanic tribes and duchies together under his rule, and forge them into a single people.” I get the impression that Charlemagne was some sort of early supporter of German unification from that, which I doubt.)
There’s also Encyclopedia Britannica. The entry on Charlemagne has this header: “Charlemagne Holy Roman Emperor [747?-814]” The article on the Holy Roman Empire puts the dates of its existence as 800-1806.
Oxford English Dictionary
Holy Roman Empire : The empire set up in western Europe following the coronation of Charlemagne as emperor in the year 800. It was created by the medieval papacy in an attempt to unite Christendom under one rule. At times the territory of the empire was extensive and included Germany, Austria, Switzerland, and parts of Italy and the Netherlands.
Charlemagne : (742–814), king of the Franks 768–814 and Holy Roman emperor (as Charles I) 800–14; Latin name Carolus Magnus; known as Charles the Great. As the first Holy Roman emperor Charlemagne promoted the arts and education, and his court became the cultural centre of the Carolingian Renaissance, the influence of which outlasted his empire.
There’s also the Merriam-Webster Dictionary’s entry, which isn’t sure: “Holy Roman Empire : an empire consisting primarily of a loose confederation of German and Italian territories under the suzerainty of an emperor and existing from the 9th or 10th century to 1806.”
I don't think that it is so much a question of traditional vs. modern but of what exactly constitutes the 'Holy Roman Empire' in your point of view.
You could take of course the year 800, when Charlemagne was crowned as 'Holy Roman Emperor'. But his Empire included about everything between the Pyrenees and River Elbe (and the Romagne), i.e. everything that would later become part of Western Franconia/France and Lotharingia.
You could also take 843, when the Carolingian Empire was divided into Western Franconia, Eastern Franconia and Lotharingia by the Treaty of Verdun with the king of the East being called Ludwig der Deutsche (Louis the German).
Then you could take 911, when with Louis the Child the last Carolingian king of Eastern Franconia died and the german Princes elected a king from within their own ranks, Konrad I.
But since neither he nor Henry I. had a very strong standing within the Empire you could take 936, when finally Otto I. was elected king of Eastern Franconia.
Or, even better, you could take 962, when he was crowned Emporer by the pope after his victory over the Magyars in the Battle of Lechfeld in 955 and the pope called him for support in Italy in 962.
Well, since everything between 800 and 962 was very transitional people usually decide to take either the first or the latter as the starting point of the 'Holy Roman Empire', and I think it's no coincidence, that Charles as well as Otto are the 2 'german' rulers to ever earn the byname 'der Große' (the Great) in german (Well, except Frederick the Great, of course, but he was prussian, not so much german, ruler).
So you could say the 'Holy Roman Empire' started in 800, and there really is some continuity. Yes, the Empire contained territory that would later get away, but the borders changed all the time and the 'Holy Roman Empire' wasn't called that for centuries anyway. Also this would give the Empire a nice whole 1000 years.
But you could also say that, no, that was the Carolingian rule, it had franconian heredity laws, the king wasn't elected and even if the borders changed that often this is a bit too much, we're talking about all of France here, so better let's take 962.
My take on this is that, since we can distinguish between the Carolingian Empire, Eastern Franconia, Holy Roman Empire, Holy Roman Empire of German Nation, German Federation and all the other german incarnations, 962 would be a good starting point for the 'Holy Roman Empire', and I think at least most german historians would agree.
The source I used is:
Scheuch, Manfred. Historischer Atlas Deutschland. Vom Frankenreich bis zur Wiedervereinigung.
(I checked Amazon, I don't think there's any translation, sorry)
Well, the source is a bit weak, but since you could find all of the facts I used in any credible history book on the matter it might be ok (?)