How to reinstall Mountain Lion on a new Mac

Press Command and R keys on boot, open Disk Utility, erase the disk and reinstall Mountain Lion. You've Internet Recovery to reinstall Mountain Lion instead of discs

Hold down command-R at startup to enter recovery mode. From there, if you want to restore to factory conditions, use Disk Utility to erase the hard drive, then quit Disk Utility and install the system. Note that installing will require the download of about 4 GB of data from Apple's servers, so be sure you can handle that much data in a reasonable amount of time before starting. Also, Apple's servers are having a problem with serving updates right now... I'm not sure whether reinstalling Mountain Lion would be affected as well, but if you can, it may be prudent to wait a day or two. It wouldn't be good to erase the hard drive and then find that you couldn't get the system re-downloaded successfully!

By the way, why do you want to reinstall? If you're having problems, there may be an easier solution that someone could recommend, if you provide details. If you're planning on selling the machine, you'll want to think about securely erasing to prevent possible theft of data that may be left on the drive.

The Mac downloads OS X Mountain Lion from the App Store. When you set up your Mac, Apple registers OS X Mountain Lion, GarageBand, iMovie and iPhoto to your Apple ID, so you can install them for free whenever you want

you're saying my Mac knows where to obtain ML automatically?

Yup. There is a hidden recovery partition on your hard drive, which is what you boot from when you hold down command-R. The system on that partition knows how to re-download ML.

Note that if something happens to the hard drive that causes the destruction of that recovery partition, you will automatically boot into internet recovery mode instead, where the machine boots from a system on Apple's servers. So no need to worry about not being able to find the disks ever again... you've just got to make sure you've got a good internet connection!

The other Macs won't be affected, so you can reinstall without any worry

I do NOT want the reinstall on this machine to impact the other two licenses I have registered.

That isn't a concern. You're allowed to install ML on all the computers that you own, for personal use, and Apple will trust you to behave appropriately. They don't limit the number of times you install or anything like that.

If you have a solid way to rename the folder (will sudo do this in Terminal) then I don't have to do the reinstall.

Well, there are a couple ways to do this, and a complete reinstall shouldn't be necessary. The first way is to rename the account itself, which is a little bit gnarly:

http://support.apple.com/kb/HT1428

I don't usually recommend that, as it requires enabling the root user (which is something to be extremely cautious of in the first place) and doing some stuff that could be fairly damaging if done slightly wrong.

The alternative is to simply go to System Preferences -> Users & Groups and create a new user with the desired name. You can then migrate data from the old user folder to the new one. See:

Transferring files from one User Account to another

Some things you'd probably want to just set up from scratch rather than trying to transfer. For example, don't try to move settings files unless absolutely necessary, and for things like e-mail, Calendar, Contacts, etc, if the data is all stored "in the cloud," just set up those with the appropriate account to re-download all the data.

Keep the old user folder around as long as you need to, and when you're finally sure all the data has been migrated successfully, you can delete it. (Note that you must have at least one admin account, so either make the new account an admin account or create a new admin account for maintenance purposes.)