Several things happening here:
First, docker commit is a code smell. It tends to be used by those creating images with a manual process, rather than automating their builds with a Dockerfile that would allow for easy recreation. If at all possible, I recommend you transition to a Dockerfile for your image creation.
Next, a docker commit will not capture changes made to a volume. And this same issue occurs if you try to update a volume with a RUN step in a Dockerfile. Both of these capture changes to the container filesystem and store those changes as a layer in the docker image, and the volumes are not part of the container filesystem. This is also visible if you run docker diff against a container. In this case, the upstream image has defined the volume in their Dockerfile:
VOLUME /var/lib/mysql
And docker does not have a command to undo a created volume from the Dockerfile. You would need to either directly modify the image definition from outside of docker (not recommended) or build your own upstream image with that step removed (recommended).
What the mysql image does provide is the ability to inject your own database creation scripts in /docker-entrypoint-initdb.d, which you can add with your own image that extends mysql, or mount as a volume. This is where you would inject your schema, or initialize from a known backup for development.
Lastly, if the goal is to have persistence, you should store your data in a volume, not by committing containers:
docker run -v mysql-data:/var/lib/mysql \
-e MYSQL_ROOT_PASSWORD=my-secret-pw -d mysql
The volume allows you to recreate the container, upgrade to a newer version of mysql when patches are released (e.g. security fixes) without losing your data.
To backup the volume this will export to a tgz:
docker run --rm -v mysql-data:/source busybox tar -cC /source . >backup.tgz
And to restore a volume, this creates one from a tgz:
docker run --rm -i -v mysql-data:/target busybox tar -xC /target <backup.tgz