How to gift someone whose iOS device is registered in another region or country?

You can't, you can only gift items to people who are in (and whose account is in) the same country as yourself, you can't gift to somebody in a different country.

You can contact 1-866-910-6558

They can assist with content/credit gifting to someone in another country. (Depending on country specifics).

They won't be able to help with gifting iTunes content to another country - http://support.apple.com/kb/HT2736 :

You can't redeem purchased gifts outside of the country of purchase. For example, an iTunes Gift purchased in France can't be redeemed in an iTunes Store outside of France.

http://store.apple.com/us/help/gifting

"When you shop for friends and family living abroad, you will be billed in the currency of the country where you want your products to be delivered. For example, if you shop the Apple Online Store Mexico, you will be billed in pesos. Apple does not charge a fee for international orders, however, your bank will determine exchange rates when the order is shipped and your credit card is charged. Any additional charges for an international transaction are determined by banking partners and the issuer of your credit card. Please contact your credit card issuer for details."

Am I misreading this?

If you live in the US you can purchase an item, pay with their currency, and they will get the gift and can redeem the gift card as it was paid for with the currency of the country.

That refers to the Apple online store, not the iTunes stores - you can't gift ibooks, music etc from iTunes to people in another country