I had a thought.
Who told anyone that John Connor was John Connor?
John fucking Connor did.
In DC Comics there's this time traveller called Rip Hunter, but it turns out that that's not his real name because he doesn't want any of his time travelling enemies to backstep and murder him in his crib. It's an effective move any asshole picking up bar floozies to bareback is more than aware of: Fake name = No child support.
But Sarah says "Never lie about who you are, I want them to find you, I want them to know who you are, and who I am so that they'll come after me, and then your father will save me, and then I will have you, my precious little soldier. It's very important that you say your name as often and as loudly as you can until the machines really begin to take notice."
Ipso facto Sarah killed Ginger.