It has to do with the fact that does not exist unless a and n are coprime. Consider Def 1: . That means that . This algebraic step can only be done if you require that a and n are coprime.
So technically Def 1 does NOT imply Def 2 directly. The correct statement is actually: Def 1 (Def 2 OR )
Consider a concrete example. is a Carmichael number:
and as the theory suggests.
but because 3 and 561 are not coprime.