Just who is misinformed?

I recently returned from a short trip to the USA. This trip had me connecting through Dublin. I was surprised to see Dublin had US immigration officers there so you clear immigration in Dublin and pass right through when you get to the USA.
I was travelling with my two eldest sons back to the USA so they could see their dad for Xmas. The first thing the immigration officer asked me was "where's dad?" I told her we were meeting him on the other end (Chicago). She stamped our passports and I decided to ask her if my two Brit born sons NEEDED US passports for entry to the USA. She said no as long as they have a passport of some kind. I asked if she was sure because I read that they did indeed need to enter the US on a US passport if they have an American parent. She said no they just need to fill out a different form. She held up the green immigration form. Now just who is misinformed here? All this time I believed my Brit born sons NEEDED a US passport for USA entry and now an immigration officer is telling me otherwise?

My kids have travelled back and forth for years on their UK passports-and besides the occasional immigration person saying something like 'you should get them US passports' I never had a problem.  I was really worried about the teenagers going on their own this summer, but it didn't come up even though they said they were going to visit their Grandparents. 

How confusing!