This answer is meant as an addition to the answer by Martin Kochanski and comments below it.
The angle, however, is different. Rather than focusing on historic documents and Latin as a language, I would like to bring more context in terms of history, politics, and other languages.
First of all, let me quote and comment on the original question:
"Moscovia is usually used nowadays as and old-fashioned reference to Russia"
Below I am going to explain why "Moscovia" is not just an old-fashioned reference to Russia.
"the term might originate not from any old Russian use, and not even from non-Russians incorrectly pronouncing a Russian word, but from a deliberate attempt to latinize the term"
Here there is an assumption that there was a popular (?) term used by local people to name their country and that term was latinized.
Below I will explain the origin of the word "Moscovia" and why it most likely was not an attempt at latinizing some local term.
Now to be crux of the matter.
As others have pointed out, "-ia" is a popular suffix which leads to creation of Latin words denoting "realms"/"countries": Germani → Germania, Frank → Francia. Moscovia and Russia follow the same principle: Moscow-ia, Russ-ia.
Moscovia is certainly not an alternative name for Moscow. In the olden days people called that town "Москва" (Moskva) but as the language has grammatical cases, when they wanted to say "to Moscow" they said "в Московь". "Московь" is pronounced like "Moskov" with a soft "v", not like "Moscovi".
Let's further consider that Moscow ("Москва" / Moskva) is a town, which has become the centre of a significant political power around the times of Ivan the Terrible (although the town itself is of course older). That power with its seat at Moskva as well as the lands it ruled were called Moscovia at least by some.
Given the above, it should be clear that "Moscovia" was a standard way to name a country/territory ruled by the power in Moscow. Moscow → Moscovia.
Now, how did the locals call their country?
The question may seem obvious but actually it is not. Back in those days the vast majority of people who lived there did not think in terms of countries. They thought in terms of towns and lordships. That was natural as almost everyone lived and died in just one village or town and there was no need whatsoever to even think about other lands.
So based on the above, I believe it is highly improbable that "Moscovia" was a latinization of a local term meaning the same thing.
If you disagree, I'd love to hear what that local term might have been.
Now where does the word "Russia"/"Rossija" come from?
To my best knowledge, the re-branding of the country was done under Peter the Great.
It was under his rule that efforts were made to ensure the country is called "Rossija"/"Russia" both domestically and outside of Russia.
This re-branding was important as "Russ-ia" sounds like something related to "Rus". It was politically important to tether the name of the country to the well known ancient civilisation and claim that Moscovia is not just some realm around Moscow but is actually a direct inheritor of Rus (which was an old civilisation with a golden age around 11th century). The timing of the re-branding was right - by that point "Rus" was essentially destroyed by the Golden Horde and actually Moscow itself, so there were few who could object.
So it is quite confusing when things like "The Russian city of Novgorod" are written or said. Initially, Novgorod was a princely state as a part of Kievan Rus, founded at least a century or two before Moscow, and a thousand years or so before the words "Russia"/"Rossija" became widely used. It was a town of "Rus" for sure. Novgorod was fighting wars against the Grand Duchy of Moscow as early as the 13th century, while Grand Duchy of Moscow was acting as a tax-collector for the Golden Horde, pillaging the lands of Rus, those of Novgorod Republic included. Eventually, in 1570 Novgorod fell and was invaded by Moscow(ia) in an event known as the Massacre of Novgorod. The result was that Novgorod was included into the lands of Moscovia. In Russian and Soviet textbooks the event is called a "fight for unification of Russian lands".
So, Moscow(ia)'s has been pillaging the towns of Rus - including Novgorod - and was later re-branded as Rossija/Russia, claiming to be the only descendent of Rus. The reality is that there are other modern states which are arguably more direct descendants of Rus. For example, "the mother of all Rus towns is Kyiv" and few dispute the fact. But the Kyiv is the capital of Ukraine. So that's number one.
Number two is that Novgorod is a Russian town because if was invaded by Moscovia, presently called Russia. Historically, however, it is a Rus town, which is a massive difference.
Now to the difference between "Moscovia" and "Russia".
Both words mostly refere to the same entity. Yes, "Moscovia" is seen as archaic. However, from historic and political points of view the difference is that "Moscovia" precisely names a country ruled by the power in Moscow, while "Russia" sounds as a direct descendant of "Rus", which creates a confusing and false historic narrative (yes, Russians may disagree).
The word "Moscovia" is still used nowadays by people who want to be precise and avoid calling that country in a way suggesting it is "Rus".