Speaking as a Haskell user:

Maybe chaining is just sequencing functions against results with bind.

Lifting is a general concept that is applicable to functors, applicatives, monads, etc.

--> signifies the result of the computation

    fmap (+1) [0, 1, 2] --> [1, 2, 3]
    map  (+1) [0, 1, 2] --> [1, 2, 3]
    fmap (+1) (Just 2)  --> Just 3

In the above examples, fmap (from the Functor typeclass, polymorphic) and map (monomorphic function for mapping functions over lists) are lifting the functions they are passed.

There's a little bit of regrettable redundancy in Haskell's typeclasses, fmap, liftA, and liftM (functor, applicative, monad) all the same thing.

But liftM, while relevant to monads, isn't >>=, which is bind. bind being what's relevant to Maybe chaining (just using >>= against Maybe values).