- is higher-level testing a standard term in software engineering?
No, I don’t think so. It is just a description, which distinguishes those tests from unit tests in the specific context of the book chapter where you found it.
- does higher-level testing belong to the integration test as it pins down behavior for a set of classes?
To my understanding, it is the other way round: the term “integration test”, when used in opposition to unit tests, can be seen as one kind of “higher-level” tests (among other types of tests).