I agree that this is mainly what they are interested in, names used in legal docs/proceedings. It does not mean informal names or nicknames or handles used in social media etc.
There is one more context though in which may be important to show another name even if not used in legal docs - where one is known professionally under a different name and it's relevant to the verification of who the person is/history. This is what I meant when I referred to "doing business as" (a legal term used for companies).
For example, your legal name is Nicolas Kim Coppola and you have never changed that name or used another for legal docs - but you are widely known professionally as Nicolas Cage. (Perhaps more common in some places where 'ethnic' or difficult or very long names are shortened or simplified, or extremely common names modified to distinguish from others with the same name).
Again, it's a narrow case that will likely not apply to many. But if you're publicly on the internet professionally under a different name, even if you don't use it in legal documents, you should assume that you may be internet searched and proper matching may help establish bona fides / identity
Names Hustle more quickly (and non-matching may cause confusion or delays).
I'm sure there are other uncommon cases but hopefully should be obvious.