I tried this:
[1] Used GParted on the GPT formatted 4GB drive.
[1.1] Formatted the partition in Ext4 and could not write.
[1.2] Formatted in FAT and could write.
[2] Using Disk, formatted drive as MBR, which is the default option. Tried to
create and format an Ext4 partition. Disk crashed, no partition was
recognized by GParted.
Back to [1.1] and [1.2] but on MBR this time, same behavior.
So I was wrong, in my case at least the write permission was denied because
of the Ext4 format of the partition, and not because of the format of the
disk.
So the question remains: is there an option to format in Ext4 without having
to modify the user's rights manually?