You’ve got most of it, just a little out of order.

When Homer “invents” the extra chair legs, and is applauded by the family for it, he then takes a moment and talks to his picture of Thomas Edison. He then notices that the idea for the chair legs came from his picture of Thomas Edison.

Lisa points out that it isn’t a recorded invention by Edison. So Homer and Bart set out, with the electric hammer, to the Edison museum to destroy the invention so Homer can claim it.

While exploring the museum, just before destroying the chair, Homer notices a chart/diagram similar to the one he setup in his invention workshop. While Homer was trying to catch up with Edison, he realizes Edison was trying to catch up with Da Vinci. Realizing that he and Edison are the same he decides not destroy the chair. At the same time someone checks in on the restricted part of the museum they were in and Homer leaves the hammer behind on accident.

A news report the next day highlights both the chair legs invention and the hammer invention. The news report goes on to say how successful the devices are and the money they will make.