I think this article explains a bit better what was going on.
politifact.com
In this article it says LtCol Gibson was not given the necessary approval to take a four person team to Benghazi from Tripoli for the withdrawal at the
airport. It's not technically a stand down (I think), and it wasn't for extraction at the annex.
Snipped from the article:
Here’s what Hicks told lawmakers:
"The defense attache had persuaded the Libyans to fly their C-130 to Benghazi. ... Since we had consolidated at the annex, and the Libyan government had now provided us with external security around our facilities, we wanted to send further reinforcements to Benghazi.
"We determined that Lieutenant Gibson and his team of special forces troops should go. The people in Benghazi had been fighting all night. They were tired. They were exhausted. We wanted to make sure the airport was secure for their withdrawal. As Col. Gibson and his three personnel were getting in the cars, he stopped. And he called them off and told me that he had not been authorized to go. The vehicles had to go because the flight needed to go to Tripoli, I mean, to Benghazi. Lt. Col. Gibson was furious. I had told him to go bring our people home. That's what he wanted to do. … So the plane went. I think it landed in Benghazi around 7:30."
Thankfully, they did not have more attacks at the airport.