Although common sense is now less common than in years gone by, hopefully a little common sense can still be applied to level crossings.
I would define "large, heavy, or slow vehicles" not by exact dimensions or weights, but by being of such dimensions or weights that they might reasonably either get stuck on the crossing, or being so very slow that a train might strike them, despite the crossing being clear when the very slow vehicle started crossing.
"Slow" in this context could reasonably mean not able to maintain a normal walking speed, remembering that crossings are intended to be safe for pedestrians.
Very few vehicles are unable to maintain walking speeds, but there are exceptions, some very exceptional loads move at 1MPH or less.
Many years ago there was a fatal and destructive accident at Hixon level crossing. An exceeding heavy transformer on a special transporter was being moved at much less than walking speed. WITHOUT anyone useing the telephone, this vast load moved onto the crossing, and was about half way when a train approached at line speed with fatal consequences.
The exceptional load had a police escort, and both the police officers, and the driver of the exceptional load were criticised for not using the telephone.
"Animals" would reasonably be defined as one or more large animals that are not contained or leashed.
Pet dog on lead, or animals confined in a vehicle, no need to phone.
A single well behaved horse under good control, no need to phone.
A herd of livestock, phone.
A single large animal such as a cow or imperfectly controlled horse, I would be inclined to phone in case it took fright, despite the pedanic argument that it is an animal and not animalS.
I would also use the phone if crossing on foot, but encumbered with anything beyond light luggage, shopping etc.
I recently used the single post phone when crossing on foot carrying a bed.