If it is the starter, I wonder if you can do what myself, and other subaru owners have done to fix their starter noises. I removed my starter and lubricated the shaft that the gear rides on, and the other moving parts in the starter. My start up noise was a little different than that noise, but maybe it's just because of the different starter design. I think the problem we had with our subies is that the starter stays temporarily engaged after the engine starts, due to lack of lubrication and the gear not moving back.
Something you could try before spending what looks like $150+ on a replacement starter.