Not saying you're wrong, what you're saying makes sense, but my cryptology classes describes the stages of hashed authentication the way the guy you're replying to describes things (client sends hash of password, server compares hashes).
I'm not saying what I was taught (intro level cryptology) is correct, up to date, or into the technical weeds enough to distinguish, but can you provide a source that backs up your position?
I'm very interested in this discussion and I'd like to see an authoritative source. I can pull the book I am referencing if you'd like, let me know and I'll find it.