An earlier reference than Garner–Hirschowitz's paper is Tholen's [*MacNeille completion of concrete categories with local properties*](https://core.ac.uk/download/pdf/293149837.pdf) (1979), in which these objects are called *quasi-initial* (Definition 1.1), defined as those objects that are weakly initial and *prequasi-initial* (which is exactly the automorphism condition you describe).