The idea that a peripatetic axe-murderer was active in the US, principally in the midwest, in the first years of the last century is not original to the Jameses. It was suggested as a possibility at the time, by Special Agent Matthew McClaughry of the Justice Department’s Bureau of Investigation (forerunner of the FBI), and the idea has been followed up by several other writers, of whom the most interesting are probably Beth Klingensmith, who wrote a paper on the subject, and the author of Getting the Axe, a blog run by a writer calling himself 'Inspector Winship'.

The killings discussed by the Jameses feature the same cases as those discussed by Klingensmith and Winship, but their casebook is larger (McClaughry's list of cases began only in 1911) Broadly, it's certainly true that there are significant similarities between at least some of the killings, among which we might identify the apparent illumination of the murder scene by an oil lamp that had had the chimney removed and the wick trimmed to limit the amount of light it cast in addition to the factors you list in your question.

I discussed this problem in an essay about the best-known and central case in the apparent series, the murder of a family of four and two houseguests at Villisca in Iowa in 1912. Broadly, my view is that the use of an ax in almost every case was perhaps not so remarkable in itself; while there certainly was an unusual concentration of ax killings in the Midwest at this time, almost every family in rural districts owned such an implement, and often left it lying in their yard; as such, it might be considered a weapon of convenience. Similarly, the fact that the victims died asleep in their beds was likely a consequence of the choice of weapon; an ax is nearly useless against a mobile target. Yet other similarities among the crimes are much harder to explain away. In eight of the 10 cases discussed by Klingensmith, the murder weapon was found abandoned at the scene of the crime; in as many as seven, there was a railway line nearby; in three, including Villisca, the murders took place on a Sunday night. Just as significant, perhaps, four of the cases—Paolo, Villisca, Rainier and a solitary murder that took place in Mount Pleasant, Iowa—featured killers who covered their victims’ faces, three murderers had washed at the scene, and at least five of the killers had lingered in the murder house. Perhaps most striking of all, as I mentioned above, two other homes (those of the victims of the Ellsworth and Paola murders) had been lit by lamps in which the chimney had been laid aside and the wick bent down, just as it had been at Villisca.

Whether or not all these murders really were connected remains a considerable puzzle. Some pieces of evidence fit patterns, but others do not. How, for example, might a stranger to Villisca have so uneeringly located (as reconstructions suggested he or she did) the parents' bedroom by low lamp light, ignoring the children’s rooms until the adults were safely dead? On the other hand, the use of the flat of the ax blade to strike the fatal initial blows does suggest the murderer had previous experience–any deep cut made with the sharp edge of the blade was more likely to result in the ax becoming lodged in the wound, making it far riskier to attack a sleeping couple. And the Paola murders have striking similarities with Villisca aside from the killer’s use of a carefully adapted lamp; in both cases, for example, odd incidents occurred the same night that suggest the killer may have attempted to strike twice. In Villisca, at 2.10 a.m. on the night of the murder, telephone operator Xenia Delaney heard strange footsteps approaching up the stairs, and an unknown hand tried her locked door, while in Paola, a second family was awakened in the dead of night by a sound that turned out to be a lamp chimney falling to the floor. Rising hurriedly, the occupants of that house were in time to see an unknown man escaping through a window.

All in all, then, the idea that one killer committed at least several murders at different places in this period is one thats worth taking fairly seriously. Whether this pattern extended as far, and for as long, as the Jameses suggest seems less certain, and there's an obvious danger that simply identifying unsolved axe murders across a 15-year period across half a continent will lead to "false positives" that are a product of the fact that anyone trying to kill several people at the same time with an axe will be likely to adopt a fairly similar modus operandi.