Push a button to light it up when you inadvertently cut someone off, stop too short, etc. Some percentage of road rage incidents will be nipped in the bud. (You can buy something like this right now.)
No further explanation needed. All comments welcome.
I like the concept of a Sorry light for the reasons that you mention. However, I wonder whether the insurance companies would find a way to use that to their advantage and your potential disadvantage. For example, if the Sorry light was used a certain number of times within a certain timespan, either raise your premiums or decline to insure you in the future.