Being specific is just fine. IIRC I was cautioning against your tests being too smart. Things like generating tests and test results rather than copying and pasting (usually with slight differences between tests). In your code DRY and clever are helpful, in tests it can mean that you aren't sure what the actual test is testing (since it relies on code which ITSELF might be broken to generate the test).
As to specificity, there are several levels which are normally used. Unit tests are a single class or method or function ... well, a single unit. These are tested in isolation to ensure they work. This might involve mocking out other pieces. This is call, unit testing. Then when you put those units together, sometimes with outside systems like databases, they are called integration tests. When testing a series of functionalities in a more real-world use case they are called functional or end-to-end tests. There can even be larger scope tests but they usually involve corporate process (UAT, acceptance, etc). Anyway, don't let specificity stop you.