I didn't read the book but I thought the general consensus was that James was given a kinda of ultimatum after the Six Degrees tour, he improved and mantained his role in the band. I never heard even rumors of him being at risk around Black Clouds.
Yup. Nothing like that occurred around BC&SL although sometime around that timeframe or maybe a few years earlier Mike had said online that they would not have picked James if they needed a new vocalist at that time. James was quiet about it publicly but I think its pretty clear he was hurt/pissed off by it.
For a little more detail, after the SDOIT tour Mike was not happy with James and wanted to replace him. It appears in the book that the other members and management were not as keen to replace him. Mike had some strong statements, John Myung said nothing, Jordan said he was too new to take an opinion, and John Petrucci had some very tempered statements about James. In other words, I got the sense that JP had some concerns about James but never really thought seriously about replacing him. In the book he pretty much said that James didn't seem happy in the band and was wondering if he was still into it.
Mike on the other hand said he was unhappy with James's stage presence, his interaction with the audience, his weight, his vocal performance (he had trouble consistently hitting the high notes around that period but wasn't that bad overall), and I really got the notion that Mike was ready to replace him but the other guys were not. So I think ultimatum is a strong word but they did tell him they want to see an improvement in his performance. James took it to heart and admitted that he hadn't been a good place, was depressed from his vocal chord accident in the 90s, and was ready to improve. He started taking lessons again, started working out, stopped drinking on tour (he was never a heavy drinker anyway but alcohol isn't good for the voice), stopped smoking cigars ( I was shocked when I read that he smoked), and you can see a big improvement between the 2002 tours and the 2004 tours.