Sorry to seem unhelpful. I'm just telling you what I learnt from the textbooks. As Baryshev states, his view is not the textbook view - he says it is different from Misner, Thorne & Wheeler, as well as Zeldovich & Novikov. I merely added the cite from Wald to give a more modern citation, in case you think MTW and ZN are out of date. To that one can also add Straumann, Carlip, & Donoghue. Perhaps Baryshev is right, but since it takes more time, energy and interest than I have to evaluate Baryshev, I'm just saying I'm not interested in discussing Baryshev's work at this moment. But if you are interested in reading it for yourself, I'd certainly like to hear when you are done whether you think his view, or the textbook view is right.