Hi, thanks for your questions. I'm not capable of answering both or any to the best that I can. My article is largely based on Pocock's work 'The Machiavellian Moment'. He didn't discuss anything much related to your questions. But I will try.
If I read you right, the 'biblical histories' refer to events such as the Flood, the Fall, and Genesis etc. I see these 'histories' as part of theology rather than 'human history'. They have indeed influenced our historical consciousness today. The most obvious one that I've discussed is the addition of human agency in history.
It may have also influenced many developments later on. For example, these biblical histories have long determined the length of human history (it's 6000 year if I remember right). It wasn't until modern science, geology in particular, that gave birth to the notion of 'pre-human history' or 'natural history'. It would be difficult to say if natural history would've been conceptualised without the inputs from biblical history. But since these histories preceded natural history, I would say it's plausible that they've contributed.
If you're asking if I think these biblical histories were once considered history itself, then yes. Intellectual history clearly says so.
I'm not remotely familiar with Herodotus and Thucydides. So, I cannot answer that. Sorry!