For example, hurling a fireball from an open palm is vulgar magic. Tossing a ball of pure flame, would be something pretty unbelievable to the Consensus. Even if no one was around to actively disbelieve, consensual reality would fight such an obviously intrusive event.
Coincidental magic, by contrast, is an apparently possible effect. A jet of flame shooting out from a nearby gas pipe would have the same effects on the battle, but would fit into the consensus of reality, and almost no paradox would be incurred.
Vulgar and coincidental are subjective, and often depend on the perspective of the consensus of the local area or realm. A Verbena's health potion might be incredulous in a modern western city, but might go uncontested in a rural eastern European farming village, and could even be mundane in a Verbena Shard Realm.