Improving your editing skills (one of my glaring weaknesses) are a near-perfect example of sharpening the saw. You already use an editor, now you're learning to use it faster.
I guess if we're going to play with the metaphor, learning to use a bicycle is not sharpening the saw, it's learning to use a new and better tool.
So if we go with editors, learning to type faster is sharpening the saw, just like training to walk faster. But switching to Emacs is like learning to ride a bicycle where you used to run.
In programming terms, this would be like taking the time to learn your language so that you can effortlessly interact with it and think about it without really thinking about it. Or learning your editor so you can move around your document on a subconscious level.