you are right. The axle comes out with a puller. The FSM states that when pulling the axle, I have to replace the seal and backing plate gaskets. I don't have time to do all that right now. Perhaps down the road I'll do this.
You actually don't need a puller. Remove the axle nuts, grab the drum and put it on over two of the old lugs inside out, put two lug nuts on inside out, and use the drum as a slide hammer; you just want to break the axle from the housing; don't pull the axle from the housing. Remove the lug nuts and drum. Install three of the new lugs using the steps previously provided. Beat the old lugs out and repeat previous steps if necessary.
The FSM may state that you have to replace the seal and gaskets, but the axle seal does not need replaced for only pulling the axle out far enough to install lugs. Many people here will tell you they reuse the backing plate gaskets as well. I know I've never replaced a set even after removing an axle.