Yes, it’s definitely the sort of vague wording that can be read two ways.
Although I expect that my original interpretation, 'nationwide' was probably what the journalist thought it meant, and probably what Labour think it means.
The paper version of the Guardian article also included praise for the success of LNER, for the taxpayer, not the customer. In fact PAYG is also a single leg pricing system, currently modulated by the continuing existence of the off-peak return tickets. Take those out and you have the LNER system, and PAYG getting the best fare means adding up all the single legs to get the one and only fare, the main change being that you are encouraged not to enquire about the cost in advance, just open your wallet so that the government can extract the 'full fruits of their labour' (Former Clause IV). (I assume that the 70 min train variation ticket would not find its way into PAYG.)
(
https://www.theguardian.com/busines...-rail-travel-and-will-it-make-tickets-cheaper)
PS: Saw someone else using a PAYG card at Sea Mills last Friday. He said it was the first time and he had been persuaded by the adverts. I gave him a short briefing on the perils and things to watch. Like me he was out cycling, which seems to be the main target market, because for open jaw and single journeys PAYG can save money, also good for indeterminate plans. Just need more cycle spaces to match the target market.