Anniversary is for things that are celebrated annually. And this includes birthdays, of course, but a birthday is a special case of anniversary, and you would almost never say birth anniversary, but almost always birthday.
However, unless you qualify the anniversary, it is generally taken to mean a "wedding anniversary". If I say, "Today is my anniversary," unless the context clearly indicates otherwise, I would be speaking of my wedding anniversary.
So, for other anniversaries, use a qualifier. For my employment I would possibly say "work anniversary" or something similar.
For notorious anniversaries, such as a nation's birthday, we seem to usually use a recognized proper name. For the United States, July 4 is referred to as "Independence Day." In France an important anniversary is "Bastille Day". In New Zealand, "Dominion Day."