In reply to Nick.

" I want all the templates and template parts stored in the database to be exported, probably with a button and not automatically into html files, so we can just package the Theme in zip file (which will include all the changes, new templates, etc…), and pass it along"

You can export your templates, templates in your theme and export them to a zip :) It's done using the 'export' button in the site editor

(NOTE: it's been several months since I last did this so it may have regressed since then ;))