I tried to test the blog section on my website, when I tried to add a new comment to the blog, I filled the fields of message, name and email address, after I pressed submit I saw the comment, but the name of the comment’s author was not the name when I filled before.
The name of all the comments is the value of “Display name publicly as” in the users section in the dashboard.
How can I change that the name of the comment’s author will be shown and not the anything else?
Please see: this relevant code in WordPress core. If you are logged into the site when posting a comment, the name is forced to the user name as you’ve seen in your tests.
So the answer to your question is that you need to logout before posting.