The basic answer is “they all mean the same thing so it isn’t really more or less correct to use one or the other.” These endings eventually get back to Latin in some form meaning, essentially, “referring to an origin”. (See Etymology.com for more on these endings but summarizing some points below as well.)
English Portuguese Spanish and French colonies may have used certain endings depending on all sorts of things (how literate the writer, how people pronounced things in the area, how one European country promounced another European language — The English were particularly fond of using their own rules of pronunciation so that the poem “Don Juan” would’ve been read as “Don Joo-on”. (We know this bc the poem has a certain beat that works with that pronunciation not the Italian/Spanish pronunciation).
Anyway. Language is fun.
Using -n or -ese tends to work for names ending in a vowel. Most countries fit that rule.
-ian requires you drop the last vowel so it might work better for Brazil than for Mexico. (Hence Mexican not Mexician)
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