The implication in the definition is in one direction: ka+nb+mc=0 => k=n=m=0. So yes, the definition itself says that k=n=m=0 is only necessary for ka+nb+mc=0. However, the other direction follows trivially from the axioms of the vector space: if k=n=m=0, then of course ka+nb+mc=0. The definition speaks about only one direction probably to avoid saying obvious things.

Also note that there is an implicit universal quantifier: "For all k, n and m, if ka+nb+mc=0, then k=n=m=0."