Calling it the white stolen generation doesn't exclude Aboriginals from being recognized, on the contrary it is the other way around. For example when the apology to the stolen generations was made was there any reference to white mothers or children? No there wasn't, only Aboriginals and TSI were recognized. Thus this is why the term "white" stolen generation was created to help distinguish it.
Even Aboriginals recognize the white stolen generation by the following:
After the 2008 Apology to the Stolen Generation, Indigenous leader Max Dulumunmun Harrison, an elder of the Yuin people, helped bring it out last November, when he wrote: “I applaud the Prime Minister’s apology to our mob, but what about the white stolen generation that suffered the same fate? I know many white people who went through the same pain. So why can’t this government do its healing again and apologise to the white stolen generation to bring closure to all this suffering?”[