Yes, you're wrong. There's a data link from the phone to one tower when setting up a call or similar - the ID of that tower is communicated to the core network. You can handover to another tower if that signal becomes stronger (e.g. you're moving) - this'll just give you more information, i.e. the fact that the user was moving, and you can now narrow down the location further to the parts where the two cells overlap in coverage.