The name Germany and other similarly sounding names (like Germania in Italian, An Ghearmáin in Irish and Германия in Russian) is derived from the Latin term Germania, which was first used in the third-century BC.

This is true, but "Germany" was a resurrected word from dead Latin that was not used in the English language until the 1300s, and it was not used regularly for another couple hundred years. More particularly, the word "German" to refer to a specific people and their specific language on mainland Europe didn't enter the English language until the 1550s.

Until the 1500s, the word for "German" in English was "Dutch".

isn't an entirely accurate answer, at least for English, because the word "Germany" didn't enter the English language until the 1300s, and it was not used regularly for another couple hundred years. More particularly, the word "German" didn't enter the English language until the 1550s to refer to a specific people and their specific language on mainland Europe.

The reason the words "German" and "Dutch" are used the way they are in English, and the reason the words "Deutsch" and "Nederlands" are used the way they are in German and Dutch, has a lot to do with how their respective political boundaries developed over the course of the 1500s.

Germans used to be called "Dutch" in English until the 1500s. From the late 1500s until the early 1700s or so, there was a shift in the English language where there came to be a split in the cultural identity and even the language that made a distinction between "Dutch" and "German".

The same thing happened in the German and Dutch languages at roughly the same time, only in those languages, the Germans ended up being the "Deutch" who spoke "Deutsch" and lived in "Deutschland" while the Dutch ended up being the "Nederlands" who spoke "Nederlands" and lived in the "Nederlands".

See this previous answer to this question on this sub from two weeks ago for an explanation on how the English language used to call the Germans "Dutch" but now they don't, while the Germans and the Dutch still do.