Identification of garden plant
The purple plant looks like ajuga. It is invasive in home landscapes. It was originally planted as an ornamental groundcover. It spreads by runners, but it also spreads by seeds, which can live in the soil for many years.
Don't let it flower and form seeds!
If you want to manage it, pull yearly in ornamental beds. In your lawn, we'd recommend waiting until early fall (late August) and spraying the patches with a glyphosate herbicide. This is a non selective herbicide. Only get it on the ajuga, as much as possible. After it dies, overseed (fall is the best time for seeding lawns.)