I've never read Ellen Glasgow before, so I just have to take on faith that the LoA page is correct in its comments on how her short fiction shows her transition from some kind of fiction to some other kind. I can more securely discuss the description of this ghost story as Jamesian as both totally right (it's a genteel story about a child haunting that might be real (but yeah, totally is)); and also as giving James a little too much credit in the psychological supernatural field. James merely gave us, in "Turn of the Screw," one of the best versions--and the one most often read by people in college. (It's an example of Google selection bias before Google: the more people read that story, the more likely it is to get mentioned/linked, the more people then go on to read it.)
But "Shadowy Third" probably also reminds people of James in the Wings of the Dove background: doctor marries rich woman so that he can marry his true love after he inherits all her money. It is also, like James, told with a certain amount of uncertainty and ambiguity: although the nurse narrator totally did see the ghost child, she still can't give definitive answers about what happened or what motivated the people in the story. People are as shadowy and unknowable as ghosts here. So when the narrator sums up the doctor/killer with "He was, I suppose, born to be a hero to women," we should pay attention not just to her worshipful judgment, but that "I suppose" that interrupts the sentence.
Which is probably why this isn't just a story about violence against women or ghostly revenge, but about the special ability to know and how knowledge may change one. For instance, there's lots of precise description about who is wearing what or what the garden looks like. And then we get the description of the ghost child's eyes:
For the odd thing about this look was that it was not the look of childhood at all. It was the look of profound experience, of bitter knowledge.
Oh god, how many times are you going to say "look"? If I wrote that, it would be bad style; but Glasgow focuses on the look--the look the girl gives her, the fact that the nurse can look at the child, the understanding of "look" as something more abstract than "eyes."
We see this throughout the piece, where the imaginative and sympathetic nurse gains some extra knowledge--she's the only one who can see the child-ghost besides the mother. Why? Because she's imaginative and sympathetic--a novelist manque. So she gets special information, but doesn't know how: "How the warning reached me--what invisible waves of sense perception transmitted the message--I have never known."
But there's also something special about the narrator that isn't peculiar to her, but to her profession. As her boss (and distant relative) notes, nurses get hardened by what they see and learn--just like child-ghosts. And sometimes that means that the narrator can't really fully describe what's going on, but bases her observations on some shared knowledge: "He was the sort of physician--every nurse will understand what I mean--who deals instinctively with groups instead of with individuals." That the nurse's observation about a type of doctor is directed towards a group of nurses is the sort of reflexive and ambiguous comment that keeps this from being a scary ghost story, and more a meditation on knowable-ness and tellable-ness. Like a James story.