Imagine a user votes to close a question, and the question gets closed:
closed as off-topic by Ivo Robotnik, User2, User3, User4, User5
Then, "Ivo Robotnik" changes their username:
closed as off-topic by Dr. Eggman, User2, User3, User4, User5
And then if "Dr. Eggman" deletes their account, the notice changes to the following:
As you can see, even though the first close voter changed their username to "Dr. Eggman", the name displayed after the account is deleted changes to "Ivo Robotnik", their previous name at the time they voted to close.
I came across a closed question on Hinduism Stack Exchange recently. Two out of four users who voted to close the question are no longer members of the site. Their accounts were deleted due to some reasons. But I am sure that their user names were not the ones which are shown on the close vote banner. You can find a user1195 in the comment section. User1195 is one of the users who voted to close. They commented the day before the deletion with another name which is not the name on the banner. Both the deletions happened in past 2 or 3 months.
We know that there is an issue about anonymizing deleted user's name (it's very closely related). That question says the deleted user's name is not anonymized and their name shows up on comments and close vote banners. But my question is somewhat different. The display name on close vote banner is not updated if the user changed their username before their account was deleted. The name displayed is the name at the time they voted to close the question.
Another example is a deleted question where a deleted user is involved.
Here, the user named "moonstar2001" later changed their name to "moonstar", but the displayed name in the close notice is "moonstar2001". Other users' display names are updated. So, I believe that the problem occurs only with deleted users.