The official name of Earth's natural satellite, in English, is Moon (big M). The use of the word moon (small m) as a simile is acceptable for natural satellites of other planets, but in astronomy, the official word is "satellite".
It is an American magazine (a magazine about astronomy, which makes it even worst) that came up with the awful idea that people whould write words like Moon, Sun and Galaxy, with small letters, AS IF they were not official names.
Moon is the official name, in English, of our natural satellite.
Sun (big S) is the official name of our star.
Solar system is OUR system of planets. Others are called planetary systems.
(Sol is the Latin name for our Sun)
Galaxy (big G) was the original official name of our Galaxy. Other galaxies (small g) are normally given names related to where we see them in the sky (Andromeda galaxy) or what they look like in a telescope (the Sombrero galaxy).
In the USA (mostly because of that magazine) there is now a lot of confusion about when to use capital letters and lower-case letters.
Moon share a Latin root with the word "month".
Its Latin name is Luna (hence our adjective "lunar") and its Greek name is Selene (hence the adjective selenographic).
Sun's Latin name is Sol (Solar is an adjective) and its Greek name is Helios (the heliocentric system is a system centred on the Sun).
Galaxy is a work that comes from the Greek galaktikos, meaning "milky"
(for example, galactose is an acceptable word for "lactose", the natural sugar found in milk).
Therefore, calling OUR Galaxy the "Milky Way Galaxy" is a bit redundant. The milky milk.
I continue to use "our Galaxy" (with a big G) in my articles.
The Milky Way is the part of it that we can see with our eyes: a band of faint light given off by the closest parts of the spiral arms (the stars are too faint to be seen as stars, but the total amount of light is sufficient for our brain to detect a faint band).