For anyone wondering, Deutschland, Duitsland and Tyskland all have the same latin root "theodiscus" that became "deutsch". "Allemagne" derives from a germanic tribe, the "Alemanni" who lived in modern day Germany and bordered modern day france. Niemcy and Nemetorszag both stem from old slavic "nem" (or something similar) meaning "mute". They called the Germanic tribes they interacted with mute because they couldn't understand them. "Saksa" is derived from the German region "Saxony".
The Latin root theodiscus is actually just the latinised form of the Old High German thiutisk from Indo-European teuta and means "people". Similarly, Alemanni means "all men". The Saxons were named after their typical sword or fighting knive, the Seax or Sax. It's still discussed where the term "German" originates from.
Lots of things are possible, fewer are likely, even fewer attested, even fewer attested without doubt.
Though it'd have a special irony to it: Some Roman asking a bunch of probably Austro-Bavarian or Alemanic warriors "who are you" and getting the answer "Germannen" -- "spear men". Were Italy an Island they would've said "sailors".
Probably more likely than Romans observing those tribes, noting more or less bog-standard javelins, not exactly a rare or particularly notable technology, and saying "hey let's ask them what they call javelins and add a -man after that for no apparent reason". Even if they specialised their tactics etc. around those spears would you call Greeks "Phalanx people"?
For good measure, in Arabic, Niemcy (actually “Namsa”) is what we call Austria. Germany is “Almania”, same as in French.
Of course, we have as many curveballs as you could want for European place names. Like Venice. Somehow we ended up with “Al Bunduqiya” for Venice. Have fun with that one.
Still do, Franken is on the one hand a part of Bavaria very insistent on not actually being Bavarian (and they're right), OTOH a dialect continuum reaching over the Ruhr area up to Flanders and Holland.
Charlemagne was a prince of Austrasia, just another guy in a long dynasty of conquerors but notably the one to conquer Rome (and conversion-by-genocide of the Saxons). Also a lot more Gauls back in the day in France before they went all national state and suppressed minority languages in favour of sounding more Roman.
Niemcy and Nemetorszag both stem from old slavic “nem” (or something similar) meaning “mute”.
More precisely, the first one is from Proto-Slavic root *něm- ("mute") and suffix *-ьcь ("person with that characteristic"); ě was not quite the same sound as e, so linguists mark it with the haček, just as it is marked in Czech to this day. In modern languages the nationality is called: Ru. nemec, Pol. Niemiec, Cro. Nijemac... But the country itself is called Ru. Germanija (from Latin), Pol. Niemcy (literally plural "Germans"), Cro. Njemačka (actually a feminine adjective, taken from the syntagm "Njemačka zemlja" = "German land").
The Hungarian name is a combination of the Slavic loaned *němьcь > "német" ("German") and "ország" ("land"): Németország.
Okay then, where'd Rome come up with it? Oh, "the land of the Germani," but scholars are divided on the origin of Germani. Possibly from Celt gair, "neighbor." Or maybe from a few other sources.
The word germ also comes from Latin, "sprout" but I'm not seeing anything like "people who sprout" or "from the land of sprouting seeds" or anything else to connect them at all. It's more like ger+man.
the “Alemanni” who lived in modern day Germany and bordered modern day france.
Alemans still live in Baden-Württemberg, parts of Bavaria, Switzerland and I think there's also a splattering in Austria. Basically, High Germans which aren't Austro-Bavarian (same culture, different kingdoms) are Alemans.
“Saksa” is derived from the German region “Saxony”.
That one can be a bit misleading: The modern state of Saxony has nothing to do with the Saxon tribe, the place the title was applied to slowly drifted over the ages. They're much more related to Prussians and Thuringians and even Franks, going all the way over to Flanders and Holland, than Saxons proper. (Holland, not Netherlands).
Here's a rough splotch of where Old Saxon was spoken, which you can consider "core Saxony". At least linguistics-wise it spread quite a bit especially during the days of the Hanse (becoming a minority language as far up as Estonia), but it definitely stops before arriving at a place where "goose meat" means "could you please". Utter gibberish.