I believe it would be 'Power of Attorney'. Only thing is other countries may or may not accept a US power of attorney paper. Also, if you're in a Spanish language country it needs to be in spanish. If you're concerned I'd get the proper power of attorney form for every country you are going to visit. Easiest thing is to just not become ill and go to customs & immigration yourself. A couple of times the Admiral (she's better at Spanish) just said she was the captain. I was there and just kept my mouth shut,,,, no problemo
We checked in and out of countries more than I can count but certainly more than 50 and never had a problem. As owner I was always able to go to customs and immigration. But, many places want only the captain to come ashore with passports for the entire crew. Many countries want triplicate crew lists, clearance papers along with triplicate US Documentation copies.
It can get crazy. In a couple of countries they would only accept 'COLOR' copies. We were even told where to get the color copies (probably a family member)
If you're heading out get an 'all in one' color printer on board so you can scan and print in color. Always go to shore with 3 copies of all your paperwork (crew lists, documentation and clear out papers or 'Zarpe' from your last exit etc.. Never leave a country without checking out or you could be in big trouble.
I'm sure you know all this other stuff. I'm just blabbing.