Ethnonyms are always political, so questions such as these never have one simple answer.
Latins
In contemporary American English, we do not typically use the word Latins to refer to any living people. The word Latins is most often used to refer to the Italic tribes who settled the land between the river Tiber and the promontory of Mount Circeo around 1000 BCE, who spoke Latin (not a Romance language, but Latin itself), and one group of which founded the Roman Empire. You can look through these Google Book results to see that most occurrences of Latins are in history books.
Latin
In contemporary American English, as an adjective, Latin can refer to the Latin language, the culture of the Latins (as described above), things having to do with the Catholic Church, or having to do with the people or countries who speak Romance languages "specifically : of or relating to the peoples or countries of Latin America" (https://www.merriam-webster.com/dictionary/Latin). In other words, sometimes Latin describes the people or culture of Portugal, Spain, France, Italy, Romania, but usually it refers to Latin America or Latin Americans.
Hispanic
You didn't ask about this word, but it's an important part of the discussion. Since the 1970s, Hispanic has been used in the US in official documents (and popularly as well) to mean “relating to Spanish-speaking Latin America or to people of Spanish-speaking descent” (https://www.dictionary.com/e/hispanic-vs-latino/). In other words, Hispanic describes people (or the descendants of) from Spain, or Latin America, but not from Brazil, Suriname, French Guiana, and the other Latin American countries where Spanish is not (or not as commonly) spoken.
Latino
Partially for the reason mentioned above - that it doesn't include Brazilians, etc. - in the 1990s, people in the US began to use Latino in place of Hispanic. Latino typically means "a native or inhabitant of Latin America" and perhaps even more frequently "a person of Latin American origin living in the U.S." (https://www.merriam-webster.com/dictionary/Latino). On official US documents, such as the Census, Latino has appeared alongside Hispanic since 2000.
Latinx
An even newer term, which seeks to degender the grammatically gendered Latino/a. This word began to be seen in the first decade of the 21st century and has not yet received widespread approval.
So, for more direct answers to your questions:
What would you refer to Latin people who are not Latin American as?
It depends on the context. Usually we have no reason to differentiate between the ethnic origins of different Europeans - rarely in casual conversation do we find ourselves comparing Slavs, Magyars, Celts, Germanic people, etc. with "Latins". In certain academic contexts, we could use Romance peoples or Italic peoples (though this usually refers to groups that pre-dated Romance peoples and aren't around anymore). Personally, if I were describing this group of Europeans, I might say "Romance language speakers," but more likely, I would just name all the countries.
Why is it that most non-Latin Americans are called or call themselves "Latinos or Latinas, or sometimes a Latinx, but not Latins?
This is not the case. Latino/a/x only refers to Latin Americans. Non-Latin American "Latin" people would not typically call themselves Latino/a/x.
Is the word Latin to ambiguous in non-Latin American Latin places?
See the description in the first section.
Is it considered offensive?
It's not common to refer to any living people as Latins, so it's not going to be as offensive as a recognized ethnic slur. But it is sometimes considered offensive to use the wrong ethnonym, since this suggests an ignorance about the group.
Is it not specific enough?
This is really not an issue. When we use Latin as an adjective, it simply means having to do with Latin America/ns. When we use it as a noun (which we do rarely) it refers to the Latin tribes as described above. There's no lack of specificity.
