The "categorical" definition of monads uses return and, instead of bind,

  join :: m(m a) -> m a

You don't need to specify it if you already have bind/>>= and return/pure, since they can be derived from each other.

For instance, purely by "following the types",

  f :: a -> m b
  fmap f :: m a -> m (m b)
  join (fmap f) :: m a -> m b

In other words,

  (>>=) :: m a -> (a -> m b) -> m b
  ma >>= f = join (fmap f ma)

is a perfectly good definition of bind I you have join.