Admin vs firstname and lastname
OS X uses a short user name and a long user name.

The short name is the home folder shown in /Users/xxxxx and the long name is only shown for programs and settings that know to request the long name to display. Any where you get to enter a user name, you can choose to use either the short or long name.

To see this in practice, you can either use terminal:

whoami
whoami | id


The first will print your short name and the second shows your ID and your user numbers and internal accounting.

This is covered in several Apple support articles - best being: http://support.apple.com/en-us/HT201548

I would recommend you make a back up and have someone to check your work if you try changing the short name. It can cause old backups to get erased (since the new folder gets backed up and doesn't "save space" like other Time Machine backups do) and can cause your account to not work where you'd need to reset the setup manually and create a third admin account to patch things up.