A wide diversity.
Permaculture is not about having a uniform lawn, a monocrop that is pleasing to the conformity-seeking eye. It is about inviting in all native species and letting them thrive in their natural environment, free from petrochemical herbicides, pesticides, fungicides and fertilizers. What you can do is to get some organic matter onto it, so that you can start building back the soil. If there is a local
compost seller (maybe even your town?) see about getting enough to put down 1"-2". Then apply just about every variety of seed that local seed sellers have. Even clover and alfalfa. There are some who think little white clover flowers and little yellow dandelion flowers are "weeds". They are not. They are part of the diversity of a grassland. If there are no broadleaf "weeds" in your lawn, then it is not a healthy ecosystem.
Another thing you can do is to pick any mushrooms you see and drop them on your lawn to inoculate it with their spores. Depending on how much pesticide, herbicide and fungicide was applied by the previous owners, the soil fungi may be under stress to non-existent. When fall comes, put oak leaves on your lawn and run over them with the lawn mower. Oak leaf litter is an excellent nutrient for soil building fungi, and it is a shame to burn it or bag it up and haul it off to a landfill.
Biochar is another soil amendment that is highly recommended. I don't like to apply biochar dry, as it can blow away in a slight breeze. I blend it up in water and then sprinkle, spray, or pour it onto garden beds, under trees, and over the lawn.
The thing I notice most about the pictures you posted are the bare spots. If you cover the bare spots with a combination of seed/compost/biochar/mushrooms/ground up leaf litter, they should fill in pretty quickly.