If you want to start a Notebook session or a batch run from the past using the files and data from that past period, all you have to do is select your run from the timestamp you want and then choose "Re-Run with Original Version". This will allow you to run the same code and project files with any environment configurations you have changed since the original run.
We also have a screencast demonstrating how to do this.
If you want to restart a Notebook session or batch run from the past using current project files, select your desired run by its timestamp and then choose "Re-Run with Latest Version." For batch runs, this feature will allow you run a job with the exact same input arguments while using the latest versions of your files.
If your project is associated with a git repository, both "Re-Run with Original Version" and "Re-Run with Latest Version" will pull the latest files from the configured branch. If git repository is associated with a specific commit, both "Re-Run with Original Version" and "Re-Run with Latest Version" will pull from that commit
Pro-Tip: If you really want to verify that the run uses the same previous files, notice at the bottom of the screen how the URL for the Input Files for run #3 (original notebook) and run #5 (for the re-run of #3) are the same.