Yes, I wrote a long wordy response and neglected to mention "docker start" which is a perfectly good way to come back to a stopped container after the first "docker run".
I prefer to never keep anything important in a stopped container (for very long) without committing it back to an image, and I don't like dealing with numeric ids.
Recently (it looks like you don't have this change yet) docker added the automatic naming scheme giving every container a random name of some "color_animal" pair which I think reinforces the point, stopped containers are not a place to store meaningful/persistent state information for very long.
This mishmash gets run almost every day on my docker hosts to clean up after terminated experiments:
docker ps -a|egrep -v 'ID|Up'|awk '{print $1}'|xargs docker rm
Beware, it will delete all of the stray containers you've ever created before that are now stopped!