System tests are deterministic blackbox tests that test a whole system instead of each part on its own.
System tests offer the most return of investment when writing tests since they test a wide range of code with only one test.
They are also defining a system in it's completeness while leaving the implementation unknown which enables greater flexibility in the code.
Good system tests are a better description of the system than any game design document could ever be.
A special case of system tests are E2E tests.
E2E tests test the system from the perspective of the end user in a realistic user environment.
Your E2E tests will hence represent the range of activities that your player be able to perform with the game.
While this is something E2E tests can be, they also can be implemented in a way that is hindering a good development environment/culture.
Integration tests test the interaction, the integration, of multiple systems together.
This necessitates that the tests know of the existence of system A and system B to perform the test, hence making them whitebox tests.
For that reason; Integration tests should be avoided.
If it is specific for your system, then its covered by System tests.
Else the constituents form a system on their own.
Hence you should make them their own system with a clearly defined interface to test against.
Therefore making the integration test, a normal system test again.