Elizabeth is no doubt the protagonist of this tale. However, the story maintains a kind of detachment from all of its characters, her included. As a result, the story kind of paints Elizabeth in broad brushstrokes, focusing less on her as her and more on the roles she plays as a woman, mother, and wife.
The first time we meet Elizabeth, Lawrence's language doesn't really give us a clue that this is going to be our main character; she's simply "a woman":
A woman came stooping out of the felt-covered fowl-house, half-way down the garden. She closed and padlocked the door, then drew herself erect, having brushed some bits from her white apron. (1.3)
Since there was "a woman" walking up the railway tracks just a couple of paragraphs before this (a different one, apparently), this first reference to our heroine doesn't really make her stand out to us.
In the second reference to Elizabeth, the narrator then goes on to "elaborate" that she is "a tall woman of imperious mien, handsome, with definite black eyebrows" (1.4) . . . seriously, that's all we're getting? He still doesn't give her a name—we have to wait about 30 more paragraphs for that . . .
In the third reference to Elizabeth (and many times afterwards), she is identified simply as "the mother." Well, at least it's more specific than woman? Perhaps we're getting somewhere. In any case, the title isn't exactly inaccurate; she's got two kids, John and Annie, and she's pregnant with a third.
She seems to view parenthood as kind of the ultimate role. After Walter dies, for example, she thinks to herself:
He and she were only channels through which life had flowed to issue in the children. She was a mother—but how awful she knew it now to have been a wife. (2.130)
In that moment, she seems to feel fairly detached, like her life (or at least, her life with her husband) didn't really have much meaning—but her role as a mother makes sense to her and seems to anchor her at least a tiny bit during all that uncertainty and emotional chaos. But speaking of her life as a wife . . .
Yes, you guessed, it—the narrative also refers to Elizabeth multiple times as "the wife" rather than using her name. It does not appear that she found this role particularly gratifying; to hear her tell it, a lot of her married life entailed waiting for her husband to come home drunk.
To make matters worse, once he's gone, she feels devastated by a newfound sense that she never really knew her husband at all—that they had been strangers the entire time and never knew it:
Was this what it all meant—utter, intact separateness, obscured by heat of living? In dread she turned her face away. The fact was too deadly. There had been nothing between them, and yet they had come together, exchanging their nakedness repeatedly. Each time he had taken her, they had been two isolated beings, far apart as now. He was no more responsible than she. The child was like ice in her womb. For as she looked at the dead man, her mind, cold and detached, said clearly: "Who am I? What have I been doing? I have been fighting a husband who did not exist. HE existed all the time. What wrong have I done? What was that I have been living with? (2.128)
So, yeah, things get pretty existential for her once her role as "wife" appears to be over. She then realizes that that this key relationship has basically been empty, which makes her feel pretty empty as well. Even the fetus inside her has, in her view, been transformed into a lump of ice in her belly with this realization. Yikes, poor fetus.
Given this epiphany at the end, it kind of makes sense that Lawrence would not really flesh Lizzie out too much or give her a lot of individuality. The emptiness or chilliness of her character mirrors her own devastating image of herself at the end of the story, when she feels totally cold, empty, and alienated from the life she thought she knew so well.