I just tried reproducing this in MapInfo Pro v2019.
I can see that there seems to be a problem if you edit the attributes via the Info Window. The Save button doesn't get enabled until you move the focus from the Info Window. You need to click somewhere outside the Info Window for example.
But you can't really avoid saving the data anyway. As soon as you try to Close All or Quit MapInfo Pro, MapInfo Pro will detect that there as been made some changes to the table in the Info Window. And you will be prompted to either Save or Discard these changes.
It does, however, seem that once you have edited one record and move onto another and then hit Ctrl+S to save the changes, MapInfo Pro also saves the changes that you made to the second record.
In the Browser window, the Save button gets enabled as soon as you enter/change a value in a single field and then leave that field. You can leave the field using Enter or Tab on the keyboard or using the mouse.
I also found a bug that was fixed in MapInfo Pro 17.0.1: "Data is lost while editing in browser when .NET version 4.7 is installed." Make sure you are using the latest version of MapInfo Pro 17.0. That's v17.0.4.
Can you recognize the behavior above?