Is the Japanese word for Badger Anaguma or Mujina or both?

The Word used depends on the area you are in. For example, in Tochigi prefecture they call アナグマ (anaguma), タヌキ (tanuki) and タヌキ, ムジナ (mujina). Thus, there is presented a particularly complex situation over which term is correct. Though in general, アナグマ means badger, タヌキ means raccoon-dog, and 貉・狢 (mujina) is actually a very old term for referring to badgers.

So, in short they are both correct depending on the area you are in, or both incorrect depending on the area you are in.