Zero is a limit ordinal, because it is the union of its elements.
Transfinite induction has two canonical statements. The "strong" statement, $$ (\forall \alpha,\beta)((\beta<\alpha) \wedge P(\beta) \rightarrow P(\alpha))\rightarrow (\forall \alpha)P(\alpha), $$ doesn't split anything into cases. The version used most frequently in proofs says that any property preserved under unions and successors holds for all ordinals. Zero should rarely be a special case.
Also, "limit ordinals" should totally be called "colimit ordinals". The term "limit ordinal" refers to limit points in the order topology, thus excluding zero, but this is silly.