Those are definitely "henbit". It's a member of the mint family and as you've found, mints are thuggish in the garden. Unfortunately the only way to get rid of them in a lawn is to apply broadleaf herbicide (easy if you don't mind using chemicals).

Yes, that is henbit. Use a three-way herbicide that contains dicamba, mecoprop (MCPP), and 2,4-D. Find them at big box garden centers or local garden stores. Follow label directions. You will soon be as weed-free as your neighbor.