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Implicitly, stories about Black-Eyed Children are "permission stories," in which a Black-Eyed Child (or several) seeks to enter a sanctified space (like a home, or personal property in general). Giving anything unnatural such permission is inherently dangerous, and all cultures and peoples agree on this, except where preconditions have been explicitly placed in advance on the guest or visitor (Xenios to the Greeks, integral to Abrahamic ideal).
Stories where you aren't letting the Black-Eyed Children into something, well, they aren't Black-Eyed Children stories. What they are is a good question. There's a lot of urban legends and historic myths about entities rewarding you or punishing you for how you treat strangers. The Norse always maintained that Odin was a wanderer, for example, and folk myth said that he would come in many shapes on his travels, ever one-eyed, to test the people. Be an ass, get cursed, be kind and help with his request, become blessed.
Figuring out why the Children have black eyes is probably the first step to figuring out why/if the stories are connected. Hyphema can cause it with no supernatural explanation, Greys tend to be described as having black eyes, various Greek gods have been described as having "blazing eyes" with no color specified, so on.
The creepypasta-esque background and descriptions call the whole thing into question, but what if there were some number of hitch-hiking kids that had a congenital defect in the 90s? It'd be interesting to see if sightings had a pattern chronologically and geographically spiraling out from the Bethal sighting in Texas.