How do I change my name on my iCloud account?

We are talking about your first name and last name, not the e-mail address of your Apple ID., aren't we?

I just did a test of my Mac (macOS Sonoma 14.4.1): clicking on System Settings > [my name] > Personal Information > Name brought up the request of the Apple ID password, I entered it and then... nothing happened. I clicked a second time on "Name" and then a window popped up where I could modify my first name and last name.

Go to Apple ID to change your name: https://appleid.apple.com/account/manage/section/information.

The personal information section does not allow me to change the name on my Apple ID. I only have access to change Country, Language, and Reachable At information. Any other ideas?

That's odd, I can. Try to do it on a device. On an iPhone for instance, go to Settings > [your name] > Personal Information > Name.

I've also tried it on the only trusted device listed on my account (My MacBook Air) and there is no option in the system preferences to change my name.