I'm not a user of the APIs yet but taking my understanding is that the term "testrun" translates to "test cycle". I'm guessing you have to specify the test cycle(s) you want results for by adding the test cycle key as criteria to the Key parameter. If a Key isn't provided it would make sense that the API returns results for all test cycles for the project.
Your understanding is correct in terms of testrun = testcycle. However, the search requires a query which is like "give me all testcycles in the project / folder". An that's what the api docu describes. We do not know the exact test cycle id - for this reason we start the search as we want to connect our test executions with a certain test cycle and they are organized in calendar weeks.
