Are you actually sure that Plusnet cancelled the first date (the 4th), or that was the date you asked for (because it was offered) which wasn't an available date that Openreach could meet and so it was booked for the 10th, and was that with or without your agreement?
This isn't an uncommon occurrence with Fibre installs.
Whilst I can see that you said that the order was subsequently cancelled, that doesn't explain why the order wasn't completed when the OpenReach engineer turned up.
Secondly, even when Plusnet subsequently cancelled the order "because of the so-called conflict" it sounds to me that this was given initially as an excuse by an inexperienced agent, as you say, trying to blame you. And in any event if Plusnet cancelled it, so as to be able to place a new order, they had cancelled it, so any order remaining on BT's systems is entirely the foul up of BTw and Openreach, not Plusnet.
Whilst Plusnet clearly aren't innocent in all this, the crux of the problem still seems to be within BTw and Openreach.