I had to look up cashier’s check and it does not sound familiar at all. But searching a bit further it is a thing that exists, seems to be called Bank Check around here, it’s just kind of expensive to use.
It’s much more usual to pay in cash or use an account transfer (SEPA transfer) which is usually free, but with the delay of the transfer one of the parties usually takes a risk.
This year, in relation to the rule the post is about, they also forced banks that were dragging their feet to start supporting instant transfers.
I don’t ultimately know the answer to your question though. I suspect the banks have to ask you for the origin, as if you turned up with 10k in cash, but I couldn’t find anything definitive in the time I was searching around.
In the reporting I’ve seen there is a specific exception for private sales anyway. The example they give is that privately buying or selling a used car should remain possible.