Hello Jander, Welcome to the Forum.
The worst of the damage I see is just wear-and-tear from foot-traffic. Lawns can take more foot-traffic than any other grundcover but there are limits. You should decide where you want and need to have people regularly walk. Those areas are paths, not lawn, though you can just leave them as they are. Areas of the lawn that you don't want to be regularly walked on should be blocked by a border of some kind, perhaps a planting of perennials or low shrubs. The other alternative is to regularly re-sod those areas. That is what a professional turf manager would do, but I don't recommend that for a residential lawn.
The other patchy areas may be suffering from dead thatch and drainage issues.Dethatching and adding new topsoil in depressions is the first step. There seems to be a large sunken area near that Japanese Maple. You would still need fill in with new grass in many areas. Plugs or patches of sod would work.
