It still isn’t an absolute obligation, and operators can and do decline ticket acceptance in various circumstances. That isn’t regarded as acceptable by certain posters on here, but it’s the reality.
It’s hardly unreasonable for Lumo to decline to carry someone in the OP’s situation where they could simply have joined a following LNER service.
Isn't this precisely the point though? OP was given the option, leave the train at Doncaster, or pay for a new fare on Lumo. OP chose to leave at Doncaster, thus a permission to travel between Newcastle and Doncaster on the Lumo service was implied with no further penalty to be incurred. Upon reaching Kings Cross the OP was then issued with a penalty, in the form of a UFN, for a more expensive journey than had they of just remained on the Lumo service.
The more I look back over this it should be extremely simple for OP to appeal this and win, but the fact it has even reached this stage shows how little FCC (Lumo's revenue and customer service provider, along with Avanti WC, Hull Trains, and TPE), and Lumo in turn know about the railway and it's policies.
OP boarded the wrong service in error, they were given an option, remain on board and pay the fare or leave the train at the next calling point to join the correct service. They chose to leave the service to avoid further costs, but when reaching Kings Cross an overzealous Lumo staff member who decided that how it was dealt with previously was no longer suitable to them, that isn't something you can do. You can't imply travel is authorised and no further penalty then change your mind around 2 hours later when you suddenly spot them.
Ultimately, it would be quite simple for OP to present their LNER ticket for the journey and that they were stopped when leaving an LNER service. OP was given the option, pay for ticket to Kings Cross on Lumo, or be authorised travel as far as Doncaster, without penalty or further costs. This was also caused by information that was open to misinterpretion being provided by Trainsplit. If OP hasn't been issued with any additional paperwork and has only received the one letter that they have shared with us then it should be fairly open and shut.