IMHO one can tolerate such small differences in the mathematical language pretty easily (much easier than to tolerate the ambiguity of the natural language that lacks specifiers for almost any adjective, so people drive me nuts when saying "that is good/bad/beautiful" and I drive them nuts asking "good/bad for what purpose?" or "what is your definition of beauty?" in response) if one makes it a habit to explicitly state what certain words and symbols mean in the beginning of every article or assignment (for a class, you need to do it just once in a separate handout, or you can remind it five times in presentations after which everybody should get used to it).
The full standartization is, probably, impossible, because in reality it is rather convenient to assume slightly different conventions and notation in different situations. The argument about which one is the correct one is totally pointless as any argument about definition of words. One should just tell the students that when in doubt, they should ask what exactly is meant by this or that and assure them that such questions will always be answered. We all speak to each other like Humpty-Dumpty in Carrrol's "Through the looking glass" all the time and there is no way around it except asking "what exactly did you mean?" before making any strong comment or agreeing/disagreeing with the speaker.
So, I wouldn't bother too much about "international standartization" here. What I would rather like to see is sending a clear message to the students that the same word can have different meanings in different situations and that the speaker and the listener should agree on the underlying terminology before trying to convey any message.
As an anecdote that illustrates what happens if they don't, there is now a hot dispute in Russia about the following situation: On an exam a student was given a problem that literally translates as follows:
If you purchase 2 chocolate bars for 180 rubles, you receive one for free. How many chocolate bars can you receive if you have 360 rubles?
The student answered $2$ because her idea was that "receive" and "purchase" are two mutually exclusive words. The official answer was $6$ because the examiners considered these words synonyms. It cost the student a gold medal and the dispute was taken all the way up to the Ministry of Education and is still ongoing.
So, just keep in mind and make it clear to others that everyone will speak like Humpty-Dumpty no matter what even in scientific communication and that the prudent thing to do as a listener would be to take the approach he recommended to Alice, though, as we all remember, she wasn't very fond of him or his way of speaking in the end :-)