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This link suggests that you can saves edits (Currently Editor -> Save Edits) as the same time as a project.
Is it still possible to do something similar, change the option(s) so it saves edits AND the project using ArcMap?
You are using the ArcGIS 10.x for Desktop architecture while that documentation is for ArcGIS Pro.
Consequently, for the ArcMap application, you will need to manually Save Edits.
Required, but never shown