"Latin terminology" is the correct answer, because legal English uses formal language, and Latin words are, of course, very formal. "Common vocabulary" is quite informal, there is no reason to use "archaic words", and legal English rarely uses "simple sentences", but rather very complex ones.
Legal English is used in legal writing. It is formalized on logic rules which differ from the ordinary natural language. It differs from the common vocabulary, sometimes is referred to as a sublanguage.
The influence of Latin can be seen in a number of words and phrases such as ad hoc, de facto which remain in current use in legal writing.
