I've taken my kids on several cruises while DH stayed home and every time I have confirmed with
DCL that the form was NOT required in that situation. However, each time I've allowed DD even to go to WDW with another family (twice), I've given a notarized letter to them, not only to authorize travel, but also to authorize medical care, etc. in the event of an emergency - and that wasn't even outside of the country.
That said, many times when we've flown with our kids - even when both DH & I were there - TSA makes a point of asking the kids their full names and ages, and asks us politely to allow the children to answer for themselves. We've had this happen at immigration, too, when they were reviewing the kids' passports. I explained to the kids in advance that that was a possibility and that they needed to look the agent in the eye, answer confidently, not mess around, etc., and we explained what their job was, to include the roll of making sure that kids were with who they were supposed to be with!
And now for my slightly related horror story - this is a case of a child traveling without EITHER parent, so slightly off-topic, but demonstrates why the letter is SO important for aunts/uncles/grandparents/friends......My nanny (from Ecuador with US Green Card) was recently offered an all-expense paid trip to Barcelona by friends of hers. The mom (American) lives in the US and the dad (American) lives in Spain. The son (12 years old) was going to Spain to visit dad, but mom couldn't take him, so they asked my nanny to act as escort on the international flight, stay at a hotel for several days on their dime, and then escort the boy home. Awesome, right? Well, when she walked in the door after 6 days off, I asked her about her trip and she said it was the worst experience of her entire life. She spent the ENTIRE time being held in detention at the Barcelona airport! Evidently the parents didn't give her that notarized letter, and she hadn't really done her own research to see what she might need. Immigration in Spain outright accused her of bringing the boy to Spain to sell him. No one spoke fluent English to communicate directly with the boy except for our nanny. They told her to translate and asked the boy if it was true the father was waiting for him (figured they knew enough English to recognize a few words!) To our nanny's amazement, the boy answered "no"! (He later claimed he thought she had asked if he knew where the hotel was - his dad was furious with him when he found out about this, as it was pretty clear the kid didn't understand the gravity of the situation and thought he was being funny.) They finally agreed to call the boy's father on the mobile number she gave, as he was anxiously waiting just outside. After proving who he was by matching the passport names, etc., they left the boy there but refused to release our nanny. They confiscated her Green Card saying it was meaningless to them and they could keep it if they wanted, and told her they were sending her back to Ecuador! After 3 days, they finally agreed to put her on a plane to the US. They did give her the Green Card, and she said the irony was that as she sat in detention, she marked her 5-year anniversary such that she is eligible to apply for citizenship! To add insult to injury, her luggage "went missing" at the airport and took another week before they "found" it and sent it back here to her. She said she has since learned that Spain has a pretty bad illegal immigration problem associated with people from Ecuador, so that was probably working against her, too. And - she says she won't be going on any trips out of the US anytime soon, particularly for other people - at least until her citizenship comes though. She definitely learned the hard way to do her own research on a deal like that - and I'm hoping that family does something REALLY nice for her.