The reason to take a case stat is the likelihood of imminent death or severe, irreversible injury (via herniation or destruction of brain parenchyma) caused by something that can't be temporized with a bedside procedure or maximum medical therapy. These are mostly brain bleeds, like acute subdurals and epidurals, some massive chronic subdurals and some intraparenchymal hematomas; tumors causing mass effect beyond what can be treated medically; and miscellaneous things like abscesses, empyemas, and malignant edema from large strokes and trauma. We will often take acute depressed skull fractures stat as well.
Hydrocephalus caused by tumors, aneurysm rupture, etc. usually wait because we can place an EVD to temporize the situation, although placement of the EVD itself is usually a stat procedure. Spine emergencies can almost always wait.
Strokes have a standardized response algorithm with benchmarks that need to be met to maintain accreditation, like door to CT scan, door to groin puncture, door to first pass, etc.
At a medium-sized level 2 trauma center it's most likely that neurosurgery call is not divided by sub-specialty. There is probably one attending on call for all of neurosurgery, or maybe one on cranial and one on spine. Stroke call is usually separate.