I said before that it seemed most people would say "I was home alone with two small kids".
Do you think the point of the story was that she *actually* wasn't home alone, and sending the picture is an effect to prove to JH that she is home alone?
I can't think of one good reason to say "their dad" in that context. I do find it distancing, there are options for who she sent it to that wouldn't sound odd, Jonathan, my boyfriend, my fiance, but she chooses "their dad"?
I don't know if we can guess exactly what the answer is, but it really does suggest that there's something else going on behind that story and she's folding all her lies together badly as she tries to get the story out in a way that protects her but also implies the strangers are the thing that's wrong with the picture.
For instance if she wasn't alone, there was another adult there that she didn't want to tell JH about, and maybe on top of that she's trying so hard to say *something* that implies there are still two children. And remember we have someone who did stay overnight, would have been there pretty early in the morning and who didn't see any sign of Lucas. So these two things might gel together in this story.