I believe that that last bit is a myth. I've never seen real prices affected by clearing cookies. And it would be quite counterproductive for an airline to play this trick, as it would just drive (normal) people to book with a cheaper alternative airline.

The explanation for what you are seeing is that the prices shown to you are not firm prices, they're only estimates of an attributable half round-trip price to each sector. It's possible that the two websites are using different algorithms to estimate these. But I'm pretty confident that if you actually select any flight pair and get a firm price quote on the next page, it'll be the same number whichever website you're on.

After all, the firm price quote matches a parallel ITA price quote 99% of the time, and the only times that ITA quotes don't match ba.com quotes, IME, are when there are POS availability differences.