Saving notebook, excluding output cells, without losing "working notebook" output cells

This is exactly the automatic behavior when using .m ("package") files, rather than .nb (notebook) files, in the Mathematica front end. You can enter and evaluate input cells (which are Code-style cells in .m files rather than Input-style cells) as usual, and the output cells display in the .m file, as for a notebook file. The output cells never save with the package file, however.

Package files are also far superior to notebook files for version control in other ways, primarily because the code cells are saved in InputForm format, rather than in complicated Cell expressions using BoxData. They also do not save meta-data such as the file outline cache and cell change times, which complicate version control, although these features can be turned off for notebooks, too.

Basically, if you want to do version control, you should use package files, unless you need to save the output cells, or other data such as cell tags.

If you want to convert an existing notebook to a package file, just "Save As", with "Save As Type" .m file, but keep in mind that the Input cells in your .nb file must be set as initialization cells (under the Cell > Cell Properties menu) for them to be converted to Code cells in the new .m file.

Simon's answer provides a good way to save clean versions of the file one is working on, but it is not particularly easy to automate. In particular, I was looking for a way to store two copies of the notebook I was working on - one with the output for ease of reading, and another one with the output stripped to commit to source control - and this should ideally happen automatically when saving the main file.

Saving to a package file programmatically can be done as explained in this answer, and this leads to the following options code,

SetOptions[
  EvaluationNotebook[],
  NotebookEventActions->{
    {"MenuCommand","Save"}:>(
      FrontEndTokenExecute[
        FrontEnd`InputNotebook[],
        "SaveRename",
        {"CleanFile.m", "Package"}
      ];
      NotebookSave[];
    )
  }
];

which will save a clean version to CleanFile.m whenever the main file is saved. I'm not sure how much this matches the behaviour the OP was looking for, but this is probably still the best place to document it.