Xbox Live Gamertag change.

support.xbox.com/.../change-xbox-live-gamertag

You can easily change your Xbox Live gamertag. If your
gamertag was created for you during your initial Xbox Live sign-up (meaning, you did not choose your gamertag), you can change the gamertag one time for free. Additional Xbox Live gamertag changes cost extra.

The txt quoted is in the section which yes has the Windows LIVE section below, but at the top clearly states Xbox 360 Console:

"How to change your Xbox LIVE gamertag

Change your gamertag on your Xbox 360 console

You can easily change your Xbox LIVE gamertag. If your gamertag was created for you during Xbox LIVE sign-up, you can change the gamertag once for free. Additional Xbox LIVE gamertag changes cost 800 Microsoft Points.

To change your Xbox LIVE gamertag

On your Xbox console, sign in using the Xbox LIVE gamertag that you want to change.

Go to Social, and then select your gamertag.

Select Change Gamertag.

Note the Price of the change, and do one of the following:

Select Enter New Gamertag. If the gamertag you enter is already taken, you will be prompted to try another gamertag.

Select Get Suggestions to view possible gamertags.

Once you enter or select a new gamertag, select Yes, use this gamertag.

Note Click Add Microsoft Points if you don’t have enough points to complete the purchase.

Once you confirm your new gamertag, the change is reflected across Xbox LIVE. You don’t need to tell your friends that you changed your gamertag — their friends lists will be updated automatically."

Again a clear statement from Microsoft, it should be honoured not dismissed.

Here's how to change gamertags -
support.xbox.com/.../change-xbox-live-gamertag

It does cost 800 points to change your gamertag.

LegacyStudent68 - Here is a link that explains how to change your gamertag. support.xbox.com/.../change-xbox-live-gamertag

Free changes are only given if you have a randomly generated tag and not one of your choosing (in addition to only having the account for 30 days). Since you have a gamertag that you chose, the free change would not apply to you even if you were within the
30 days.

support.xbox.com/.../change-xbox-live-gamertag

You can easily change your Xbox Live gamertag.
If your gamertag was created for you during your initial Xbox Live sign-up (meaning, you did not choose your gamertag), you can change the gamertag one time for free. Additional Xbox Live gamertag changes cost extra.