From Bug to Improvement, because this plugin only considers parameters: it allows the user to rebuild a parametrized build without entering the parameters again.

Ah, so I did misunderstand the purpose of the plugin?

Weird - I could have sworn that it used to work the way I described, but I'm obviously imagining things.

Is there any way to achieve what I want right now (re-run the same job, using the same commit), without having to resort to dummy commits to kick it off again?

I occasionally have to do this - if I encounter an error in our build scripts, for example.

I think there is no plugin with that feature, but I'd suggest you a workaround in your case.

In order not to change your job configuration disabling SCM configuration, you could create another job using its workspace:

• Create another job
• General > Advanced... > Use custom workspace:
  Directory: $JENKINS_HOME/jobs/<under inspection job name>/workspace

It could be a parametrized job in which you provide the job name you want to debug (e.g. parameter with name JOB_NAME_UNDER_TEST):
Directory: $JENKINS_HOME/jobs/$JOB_NAME_UNDER_TEST/workspace

I'd also like to have the functionality Sam Deane requested.

Can we have additional checkbox "rebuild the same commit" in the rebuild page, unchecked by default to maintain backward compatibility?

To be frank, rebuilding on the commit of the job that the Rebuild link was clicked from feels like a basic requirement of calling the functionality that this plugin provides rebuild.

Maybe the link that this plugin provides in a job's menu should be Re-use parameters or something.  But unless using the same commit is included in the functionality, Rebuild probably deceives more people than it doesn't.