I simply can't add anything, you're 100% right...
...but I have to
As Leoreth mentioned, there was nothing like "Germans" until the 1900th century. In central Europe there was from 10th century up to 1806 a federal union of independant states ruled by an elected leader (the emperor of the HRE), called first the "Holy Empire" (sacrum imperium), later the "Holy Roman Empire" (sacrum romanum imperium) and since ~1500 "Holy Roman Empire of the German nation" (sacrum romanum imperium nationis germanicæ). The name affix "of the German nation" was only added, because most of the people spoke German dialects. But still now there are many different dialects in Germany. Someone from southern Germany has REALLY problems understanding someone from northern Germany, and vice versa. This was yet very much harder in the middle ages. There was nothing like even a German language until Martin Luther translated the Bible in the 1600th century, trying to write in an understated form of his own dialect.
The HRE was ruled by an elected emperor, but usually the son of the late emperor was elected, so in fact, there were dynasties, like the Staufers or Habsburgs. Interesting fact: emperor Friedrich II. of Staufen (Castle Hohenstaufen is near Stuttgart in southern Germany) was born in Palermo in the 12th century. Even though he spoke about 5 languages, he couldn't speak German.
So there were some dynasties ruling the HRE. The most important dynastie was the Habsburgs, ruling for about 400 years until the HRE was dissolved in 1806 due to Napoleon Bonaparte's pressure. There hasn't really been another dynastie ruling for more than roughly 100 years, so it is obvious to see the Habsburgs as THE emperors of the HRE per se.
Another interesting fact: Castle Habsburg was located in (today's) Switzerland, not even close to former and today's Austria.
Prussia raised in the late 17th century in the Baltics, between Russia and Poland, Prussia was even a part of Poland. Prussia has nothing really to do with the HRE / Germany until the late 1800th century. It wasn't part of the HRE. It's only attachment to the HRE was, that the monarchs of Brandenburg (which was part of the HRE) since the 1600th century also were Prussian Dukes, since 1701 Prussian Kings . The culture, society and way of life of the Prussians differs from the other "German" nations (Kingdom of bavaria, kd. of saxony, kd. of württemberg, ...) in so many ways...the only thing in common was the German language, which only was spoken in Prussia, because Brandenburg was part of Prussia (or rather vice versa) and the Hohenzollern dynastie origins in southern Germany (again, nearby Stuttgart) and Prussia conquered several German states.
Citation from Mirabeau about Prussia: "Other states have an army, in Prussia, the army has a state.", compared to the almost "dolce vita-like" way of life of the Austrians.
Interesting fact: one of the most important monarchs of Prussia, king Friedrich II., "the great", spoke French, and not German.
Franz II. of Habsburg formed in 1804 the "Kaiserreich Österreich", "Austrian Empire" and resigned as Holy-Roman emperor in 1806. So Austria is de facto the replacement of the HRE.
In 1871, Prussia declared the Prussian king, Wilhelm I. of Hohenzollern to the German Emperor in Versailles, after defeating the French (the so-called "hereditary enemy"

).
So, within this war from 1870/1871, the first REAL "German nation" was founded.
Also, the "German emperor" had NOTHING to do with the "Holy-roman emperor". There was NOTHING like a "first reich" or a "second reich". In fact, the HRE was in the tradition of the "4-Realms" according to Prophet Daniel in the Bible, considered as the continuing (ancient) Roman Empire, the "4th realm". The Prophet Daniel dreamt, that after the fall of the 4th realm, the Devil would rise and rule our earth.
The terms "first reich" and "second reich" were just pathetic efforts of the Nazi-régime to legitimate their rule in some way.
Therefor it is simply logical to have the HRE spawn in ~800ad, rename it in 1800ad to "Austrian(-hungary) Empire" and have the Prussian nation spawning in 1700ad, renaming 1870 to "German Empire".
There is absolutely no other historical way of represent the actions in central Europe.
(sorry for the "essay", but I couldn't help...

)