Not sure about negotiating, most of the time the title has some indication of the scope and responsibility of the role. So if you are not the Head of something it’s hard for the hm giving you such title. But sometimes there is room to discuss the exact title.
This doesn’t exist at any of the big companies. There is PM, Sr., and Director. I think head, lead, group, etc., while relevant to the scope of responsibility, are made up by the individual or their team.