Bat-Girl or Flamebird, what's in a name?

After the introduction of a sidekick for Batman in the form of Robin, it was only a matter of time before others would replicate his success.

Betty Kane took to her aunt Kathy's example and adopted the identity of Bat-Girl to complement Batwoman in garnering the attention of the Dynamic Duo. After brief careers, she vanished as did auntie, aside from a brief appearance alongside the Teen Titans west team that soon disbanded.

Then the Crisis on Infinite Earths came and Betty was erased from existence, replaced by an Elizabeth Bette Kane who followed Robin's example and used a new identity of Flamebird to attract her man, once more to no avail.

When another Crisis effected the merged Earth, it was revealed that Bette was a native alongside the Justice Society of Earth-Two. Hence she is unique as one of the few, as well as Wildcat, who took over the dominant position on the new merged Earth (wherein most merged characters were originally from Earth-One with their counterparts erased from reality).

During this adventure, it was established, then, that both Earths One and Two had their own Bat-Girl... with slightly different names. And, wherein, the golden age version retired for good, since no Teen Titans existed on her world as had that of her silver age version. Regardless, both lasses were mere precursors to the dominant female protectors of Gotham City in Batgirl and Huntress.