I think your reaction was natural, and virtually all of us would do the same (unless we have a 'confidant' on speed dial).
But you raise a valid point.
If you genuinely need to be rescued, then at some point you are going to have to live with the consequences of calling for help, or die.
Given the choice, a red face, seems better than death.
Moral - make sure you can escape next time !
Daffy