It's a quirk of XP. The user setup is flawed. If, as an allegedly prudent user would, you name the administrator account by a name other than the default 'Administrator', the whole log in/log out UI goes to the dogs, unless you give it the old skool 2K look (the 'scruffy' but practical drop downs) and rather to continue to use the flashy, but flexibility free, XP buttons.
Best solution is, restore the original administrator login and forget the personalised login option. Two ways to do this should work.
1,Go to the User Accounts application in the Control Panel. From the "Pick a task" list, click "Change the way users log on or off." Clear the "Use the Welcome screen" check box (this action automatically disables Fast User Switching as well) and click the Apply Options button. Then, log on to the system using the classic logon screen, and you'll be able to enter any account name you want to. When you open the User Accounts applet in the Control Panel, you'll see that the Administrator account is now visible. By logging into this, you should be able to delete the 'June' acct, which you couldn't do when logged in on it as you'd be attempting to delete what the OS would see as the only admin acct.
2, Erm, actually, I can't remember how it's done right now, but it's a registry change to force the admin acct to be shown as a login option even whilst using the XP style login screen. It's slicker than the above, not that that's too tricky, but if it sound like what you need, I'll try harder to remember and post accordingly.
Oh, BTW, don't worry about the prudence of logging in as administrator. If someone wishes to hack your machine, the difference between an account named 'June' or 'Administrator' won't make the difference. The latter keeps things simple for you however.