Apple - Remove my full name from startup screen

You can change the Full Name for a given user account. From Apple:

• Go to Apple menu > System Preferences, then click Users & Groups.
• If the lock icon in the corner of the window is locked, click the icon to unlock it. Enter the admin name and password, if prompted.
• From the list of users, open a shortcut menu by Control+clicking the user that you want to rename. Choose "Advanced Options" from the shortcut menu.
• Change the name in the "Full name" field, then click OK.

Changing the Full Name here will change the name shown on the login screen you described. DO NOT change the short name (or anything else for that matter). Read the warning if you are tempted to change anything other than the Full Name:

Given this warning I would highly recommend making a backup before performing any changes.

I can confirm that updating the Full Name (and nothing else) here will not prevent you from logging in. However, there is always the chance that this will have other unintended consequences. If needed, you could take these steps again to revert back to the accounts original Full Name.