Well there could be more phases like performance, security but lets say you are correct. The Blackbox would certainly be in acceptance and system while whitebox would be at the unit test phase. For integration it could be a bit of both or either.
Unit testing (testing the smallest bits)
Element/module testing (testing the smallest bits that make functional sense to a non-programmer)
Integration testing in the small (putting 'bits' together and testing)
Integration testing in the large (putting all or nearly all the 'bits' together and testing)
System testing (testing all the bits on together on the intended target platform i.e. the intended version of windows, server setup, any other systems that run along side, a few of the admin and operator staff that will use it)
Acceptance testing (should be a formality)
This is a big topic - maybe worth investing in a book.