Can this also mean "what do you want to say", or would you require a "a" or "di", etc. before "dire"?
Am pretty sure I've heard this, or maybe more polite variant "cosa vuole dire" used in a few intense Italian arguments.
Exceptional! There are two translations given : "What do you mean?" and "What do you want to say", but "What do you mean to say" loses a heart....
Voler dire only translates to 'mean' as an alternative to 'want to say'. Its subject can be a person or a word / phrase / text.
Your last example needs intendere or synonym. Other cases need significare or synonym. The context decides.
"what do you mean? and what do you want to say?" are not the same thing=cosa vuoi dire?
I'm thinking the same thing. To me, "What do you mean" implies that you've already said something but it requires further clarification. "What do you want to say" implies that you have not yet said something but you would like to.
I suspect it is along the lines of, especially in arguments, "What do you really mean, what do you really want to say?" Where someone has said something possibly ambiguous, possibly offensive, and someone is asking for clarity. Or raising the level of the argument.
I found several places that said it is regional. Here is one https://dante-learning.com/eng/che-cosa-or-che-cosa-the-italian-way-to-say-what-podcast/
Here is a somewhat related post https://www.duolingo.com/comment/3320936/Qual-vs-Cosa