Further links within that main article explain how to change your payment information to 'None' on an existing Apple ID.

You don't have to create a 'new' Apple ID to change your payment information to 'None'.

If you already have an Apple ID:

When you use Apple services, such as the Apple Online Store or registering your product online, you create an Apple ID. You can use the same Apple ID for iTunes, the Mac App Store, iCloud, iMessage, FaceTime, Game Center, and more.

It is recommended to use only one Apple ID.

If you have registered your product or used any Apple services, you should follow this article to use an existing Apple ID with the iTunes Store or Mac App Store.

If you have not created an Apple ID yet, this article will assist you in doing so on your computer in iTunes or on an iOS device.

http://support.apple.com/kb/HT2589

Use your Apple ID with the iTunes Store

1. Open the latest version of iTunes.
2. Choose Store > Sign In.
3. Enter your Apple ID and password and click Sign In.
4. Click Review when asked to review your information.
5. Enter your billing information.
6. Click Continue after you enter your credit card and billing information. You can remove or edit your credit card information at any time. Learn more about updating your iTunes Store account information.

http://support.apple.com/kb/ht1918

Change your payment information on a Mac or PC

1. Click to open your account in iTunes (you may need to sign in with your Apple ID), or follow these steps:
   1. Open iTunes and click iTunes Store.
   2. Click Sign In and enter your Apple ID.
      
   3. Click your Apple ID and select Account.
      
2. From the Account Information page, click Edit to the right of Payment Type.
   
   You can see the payment methods that the iTunes Store accepts in the Payment Type section. If you want to remove the payment method, select None in the Payment Type section. See what to do if None isn't showing or you can't select it.
3. Change your information and click Done.
   After you change your payment information, the iTunes Store places an authorization hold on your credit card.

http://support.apple.com/kb/TS5366

Why can’t I select None when I edit my payment information?

Symptoms

When you try to edit your payment information in the iTunes Store, App Store, iBooks Store, or Mac App Store, you may see these symptoms:

• When you try to remove the payment information from your account, the None option is hidden.
• When you create a new store account, you're asked to enter a credit card and can't select None.
• When you sign in to the store, you must verify your payment information and enter a credit card to save your payment information.
• When you change the country or region of an existing Apple ID, you're asked to enter a credit card and can't select None.

 

Resolution

If you want to select None and remove the payment information or credit card from your account, follow these steps.

Using an existing iTunes Store or App Store account

If you have an unpaid balance or payment due, you can't select None as your payment type. To see which order you need to pay for, view your purchase history. When you sign in to the store, you may see a message that there was a billing issue with your last order. Click the Billing Info button to see the order.

After you pay for the order, you can change your payment information to None.

 

Going through the articles will eventually get you to what you need.

Enter the code located on the top of an iTunes gift card. That is what I did and it works just fine!