So you can see, just as you said, LR uses EXIF:DateTime (or XMP:ModifyDate) for Capture Date (the red rectangle at the right, French format) and (incorrectly) shows file system's last-modification date for the Grid (left red rectangle, European format). Clearly, they are not the same.
Bug, right?
Btw, do you know a good tool to edit EXIF (pyexiftoolgui cannot add Capture date, seemingly).