A language L is complete for a complexity class C if it is inC and all languages in C can be reduced to L. "Turing" is not a complexity class, so "Turing Complete" is nonsense. And if it did mean something, it would probably refer to a recursive language to which all other recursive languages could be reduced.
"Turing Equivalent" is something a programming language can be and doesn't have much to do with complexity theory.
"Turing" is not a complexity class, so "Turing Complete" is nonsense.
Good thing that complexity theory isn't the only part of CS that uses the notion of completeness. Turing (aka recursively enumerable functions) is a computability class, and it makes sense to talk about models of computation complete for that class.