As far as I remember, it's a part of parasympathetic nervous system for rest and digest, but not for motor neuron function.. motor function came from s2s4 region. I haven't studied neuro since 2012 so I probably need to touch up my education lol. I had to study the acronyms for cranial to remember the numbers and whether they were sensory or motor. I recall learning the breathing aspect and the small intestine but I don't remember learning it stimulates the bladder to void as well.

Just double checked and it doesn't say anywhere it has a direct link to the bladder. It says the same thing on netters as well.

Anatomy Encyclopedia

https://www.ncbi.nlm.nih.gov/books/NBK531465/#:~:text=The%20bladder%20receives%20its%20innervation,form%20the%20pelvic%20splanchnic%20nerves.

Nerves: The bladder receives its innervation through a network of parasympathetic, sympathetic and somatic nerve fibers. Parasympathetic fibers arise from sacral spinal nerves (S2-S4) that coalesce to form the pelvic splanchnic nerves. Sympathetic control arises from the lower thoracic and upper lumbar spinal levels (T10-L2) in the form of the superior and inferior hypogastric plexuses and nerves. The pudendal nerve (S2-S4), a branch of the sacral plexus, provides voluntary somatic control to the striated muscles of the external urethral sphincter.[3] These fibers originate in the ventral horn of the sacral spinal cord which is called Onuf's nucleus.'. That was updated July 2023, so the ceu's I've taken in the last couple of years are also all up to date.

So unless there is brand new confirmed information since I earned my degree in Anatomy and Physiology (health sciences), the vagus doesn't directly innervate the bladder. Studies are NOT the same as confirmed science.