A couple of issues here...
As to the check-in part.. To me this all hinges on what's their formal rule and what time did this all happen..
In most cases the "deadline" for airlines means you are DONE completely with ALL the parts of check-in.. That includes having you physically checked-in, boarding passes issued, *and* your bags weighed/tagged and accepted..
So, to me, IF you didn't get all this done, before the posted deadline, then I think BA would be right.. but again, I'd like to know exactly how does BA define this by not only task, but timeline.
The next part about BA (va the 0800 line) allegedly telling the OP that BA would reimburse them for the funds paid for the date change.. IF that what was in fact told to the OP by BA staff, then I think BA has some level of ownership over that answer..
This part is hard to assess as we'd need to hear the *whole* conversation to know.. I can see it as being exactly as the OP notes.. but.... I can *also* see this as being a misinterpretation or the like.. So, on this part, as I see it, specifics do matter.
The part about 2010 travel to me is kind of background noise and not really germane this *this* case.. For the OP it's pretty clear that they've used it (and perhaps understandably so) to form a "it's always like this with BA" opinion.. but the reality is that from a factual point of view, the 2010 case has no real bearing on the current case as the OP presents it.