And, actually, the way I edit is a sort of hybrid of the two methods. The beginning of my careful edit starts by listening carefully to the sound on my timeline.
My interview subject is a good speaker, but like most people, she does say "um" every once in a while. One of those times starts at just about 3 seconds into the piece.
Now, I can find the beginning and end of the "um" by moving my little cursor over the timeline track slowly and listening for it. But I can also look for it. To do that, I have to expand the audio track out so that I can see the waveform of her speaking. I do that by clicking the little right-facing arrow next to "Audio 4" in my timeline.
Now, I can actually look for the waveform (also I zoom in using the slider with the little mountains on it), and place my cursor exactly where her "um" begins. Then I select the razor tool at the top of my window.
My mouse's cursor takes on the shape of the razor tool, and I then click on the timeline's video or audio to cut it. This is what it looks like after I have cut in the position of my cursor.
Note: You can also just place your cursor wherever you want to cut and hit "command - k" on your keyboard.