It has many names, depends on who you ask. In English, one name for our moon is simply the Moon. Notice the article the and the capitalization making it a proper noun. This is similar in convention as the Galaxy for our own galaxy that is otherwise also known as the Milky Way. But the word galaxy actually already implies something related to milk in Greek anyway (galakt / γάλακτ = milk), so some linguistic purist frown upon the use the Milky Way galaxy as redundant.
Another name for our moon could be Luna by the ancient Roman deity that is, according to ancient beliefs, embodiment of the Moon. Roman / Latin names for celestial bodies (and sometimes their adjectives) are frequently used. In fact, we all know e.g. the planets Mars, Neptune, Venus,... by their Latin names for a god of war, god of fresh waters and the sea, and goddess of love, respectively, and less frequently use their Greek equivalents Ares, Poseidon, and Aphrodite.
And the Moon / Luna is known by many other names in other cultures. For example, in Greek mythology, it would be Selene, goddess of the Moon, or rarer, Cynthia or even Artemis, both the names of the Greek goddess of the hunt, forests and hills, the Moon, and archery. Obviously, this implies another name for the Moon - Diana, the Roman goddess of the hunt, the moon and birthing. And, say, Chinese equivalent would be Chang'e, their mythological goddess of the Moon (relevant to space exploration since the China National Space Administration chose this name for their lunar exploration program, and you might have heard of some of the Chang'e lunar orbiters and landers, e.g. the Chang'e 3 that delivered a small rover named Yutu, in English Jade rabbit, to the Moon). The list could go on, as evidenced for example by this Wikipedia's list of lunar deities.
So, when it comes to terminology and naming conventions (nomenclature), you should never neglect to mention what language, culture and use you're interested in. I know you did, I'm just trying to additionally emphasize that it makes a difference. But, to cut the long story short and answer your question more directly:
Does The Moon have an agreed upon international astronomical name?
Yes. The Moon. And this is the English language naming convention recommended by the International Astronomical Union (IAU):
The IAU formally recommends that the initial letters of the names of
individual astronomical objects should be printed as capitals; e.g., Earth, Sun, Moon, etc. "The Earth's
equator" and "Earth is a planet in the Solar System" are examples of
correct spelling according to these rules.
Additionally, if I may, what you posit in the penultimate paragraph of your question seems somewhat flawed. There is a distinction between a sun and the Sun in that while they both suggest a star that is a parent to planets, i.e. a planetary system and not merely a star, the former form is both frequently used (saying, e.g., "This planet has two suns because it orbits a binary star." would be perfectly acceptable, why not?) as well as both being distinct enough with the use of articles a or the, so they cannot be ambiguous with proper use.
If it helps make a difference, here's a quote from Curious About Astronomy post on What are the names of the earth, moon, sun, and solar system?:
The name of our planet is the Earth. The name of our moon is the Moon.
The name of our solar system is the Solar System.
Notice that I capitalize them, because when used as names, they are
proper nouns. This also helps us distinguish between the planet Earth
and earth (meaning soil), between the Earth's Moon and moon (meaning
the natural satellite of a planet), and between our Solar System and
any other solar systems (since any system containing a star and a
planet or a planet-forming disk can be called a solar system.)
The Wikipedia page you quote seems only partially correct and neglects to mention capitalization of proper nouns. Which is somewhat amusing considering its own page on proper nouns clearly stating:
In English and many other languages, proper names and words derived
from them are associated with capitalization;