Thinking back to the movie, various books, and my own training.
I think Danger Close is the call to cover dropping stuff on your own position. If you are having to shell your own position you likely aren’t observing niceties of normal call for fire procedures so whether that call is made or if it is more “I need mortars on my coordinates, NOW!” is open.
“Broken Arrow” was more a general call that a unit was in danger and needed priority on all available air support. Col Moore wasn’t calling for fire on his positions, he needed aircraft to pound the enemy forces that were engaged or moving to engage him. He did not want ordinance dropped on his position and the only strike that hit his position was the friendly fire incident as depicted in the movie. These would have been considered “danger close” and that specific call would be made to alert everyone friendly troops were near the strike zones.
There’s another line after that, Final Protective Fire (FPF) that is machine gun and mortar fire right at the edges of your own lines (say within 100 meters) to provide another curtain of bullets and shrapnel the enemy has to get through. We were taught those were all pre-planned fire coordinates with your artillery and mortar support. In addition, you’d give the platoon machine gunners pre-assigned sectors to sweep with fire regardless of seeing an enemy. That call is something like “Fire FPF!”
After that, there’s the “it’s all gone to hell plan.” In the movie “We Were Soldiers,” that’s what B company 2nd platoon (the troops who chased the Vietnamese soldier into the brush and got cut off) has in place when they are essentially being overrun at night by Vietnamese soldiers moving through their position looking for them. It’s the last chance roll of the dice. With luck, you are dug-in (even a hasty, 6 inch deep position is going to help tremendously) or at least prone. People standing and moving make much better shrapnel catchers than folks hugging dirt so while your odds aren’t great, they are a hell of a lot better than not calling in fire.