Safety valve can be characterized by full-opening pop action and relief valve which opens further with the increase
in pressure over the opening pressure. What is the popping point tolerance for both types of valves?

Refer to ANSI API 576 and 510 Inspection of Pressure relieving Devices.
The recommended practice describes the inspection and repair practices for automatic pressure-relieving devices commonly used in the oil and petrochemical industries.
This is just a guideline. there might be state codes for high pressure vessels and the popping point tolerances.

http://www.api.org/certifications/icp/programs/upload/510BOK0607-3.pdf

There are two different valves. One is a safety vale and one is a relief valve.

Safety Valve: An automatic pressure relieving device actuated by
the static pressure upstream of the valve and characterized by full-opening
pop action. It is used for gas or vapor service.

Relief Valve: An automatic pressure relieving device actuated by the
static pressure upstream of the valve which opens further with the increase
in pressure over the opening pressure. It is used primarily for liquid
service.