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You can enable editor tracking:
You can also enable this with hosted feature services:
The original screenshot you sent is small, so it's tough to see what's going on. It looks like edits are being made, and then they are labeled with the editor. Is that correct? Is this what you are looking to accomplish?
I think getEditSummary() only returns the very last operation performed.
Just to be clear: you are seeking a method to expose the full history of edits?
If you are working with an enterprise geodatabase you can accomplish this through archiving. You can publish the Historical Archive feature class as a service. You will then have access to all of the edit history when you query (i.e. identify) the layer in your application.