Correct, you are waiting for the RemoveLoginAsync to return before you execute the next line of code. However, there is benefit to doing this...
While the RemoveLoginAsync is executing, it won't lock the thread (because it's async). So with the second example for instance, if you have 4 threads available for your app, and 4 requests come in, all waiting on a synchronous call, then additional requests will enter a queue before they can be processed.
But if you have an async call while waiting, then it frees up that thread to process more requests.