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?
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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.