You're thinking of the wrong Roman empire.

So after the fall of the western Rome the Carolingian empire rose up in the west and under the rule of Charlemagne the great it conquered a good portion of Italy and had the Pope declare them Emperor of Rome or alternatively according to Monumenta Germaniae Historica, Karolus serenissimus Augustus a Deo coronatus magnus pacificus imperator Romanum gubernans imperium. His kingdom then became a major centre of culture and learning in Catholic Europe. Later the Caroligian empire was split into a number of Rump states on the death of Charles The Fat. One of which was East Francia.

East Francia was kind of the Germanic speaking sections of the Carolingian empire to simplify it a bit. When the King of Francia/Germany, as it was also known, Otto 1st invaded Italy in a repeat of Charlemagne the Great he first drove the Lombards out of Northern Italy declaring himself also the King of Italy. Later pushed further south taking control of Rome and gaining influence with the Pope. This combined with the fact he was well respected for his part in defeating the Hungarians and that he had just marched into Rome with a massive army lead to the Pope to crown him Emperor of Rome. Otto then formed the Holy Roman Empire from his territories in Italy and the Kingdom of Germany unifying it into a single polity although the King of Germany would still be used to a certain extent going onwards.

So you see Reich specifically refers to German Empires or perhaps united/greater German states.