Ok, so, the reason logs are not updated when trying to start the application is because the logs of the original database are actually written ... by the clone..... 
....whaaaat?
If opening the logs folder from setting, the name is different, but i guess the reference (shortcut??) is pointing to the same folder, as the content, the files, are identical in "both" folders :
Clone: Trying to open C:...\AppData\Local\Neo4j\Relate\Data\dbmss\dbms-3c4a8ac3-a8bc-4eb0-8629-5e8389e937b8\logs directory
vs
Original: Trying to open C:....Neo4jDesktop\neo4jDatabases\database-ebd9c666-56f7-49ea-b484-0c6b01a8259c\installation-4.1.1\logs directory
I guess it worked on Linux, but it does not on Windows.