Is this article right about redundant phrases that should be avoided?

We got an email at work from our supervisor about how to improve the clarity of our emails. One part of it pointed to this list of redundant phrases I am not sure I go along with all of them. Is it really better to say “I will write what you tell me” than “I will write down what you tell me”. To my ear, the second sounds better. Sometimes redundancy is used for emphasis.