There is no mandate for open bonnet in ASME I. Its an application/manufacturer driven thing. The use (design) of an open bonnet, is to keep the temperature away from the spring. Having excess heat acting on the spring will reduce its forces and make the valve open less than set pressure.
Note that there is a separate forum for safety relief valves.
*** Per ISO-4126, the generic term 'Safety Valve' is used regardless of application or design ***
*** 'Pressure-relief Valve' is the equivalent ASME/API term ***