It took me some work to discover what dosfsck was very explicitly telling me. He said:
Differences have been encountered between the boot sector and its backup.
He's saying: byte at offset 65 is 0x01 in original, 0x00 at backup.
Offset 65 of the boot sector corresponds to the BPB in a FAT32 filesystem. Wikipedia says:
In Windows NT bit 0 is a dirty flag to request chkdsk at boot time. bit 1 requests surface scan too.
And there really were problems in the device (the free clusters summary or something like that).
********** What I finally did ************
The DVD player is currently accepting the memory stick. This is the story: I had never [c]fdisked the stick. I had directly ran 'mkdosfs -F 32 -n <label> -v -I /dev/sda'. First I had tried mke2fs but the player rejected the stick. I thought, then, "every thing is M$ compatible" and fell back on mkdosfs. Notice how, after running that command, the device has no partitions and, therefor, no MBR. First sector on device is straightforwardly the boot sector.
Under these conditions, the DVD player I had before, did not object the stick. Then I replaced the player and, in the meantime, the stick file system was corrupted. When I tried the stick with my new (but actually perhaps an older generation DVD player) player, the latter did not like it. Second step: I run dosfsck (these two programs belong to the dosfstools slackware package) on the stick, who detects the fs inconsistency. Dosfsck then fixes (supposedly) it. But still the player dislikes it. (My machine never complained.)
Finally, not caring about my data being lost this way, I proceed to reformat. This time I intend to make things such that the stick is left as it was when it left the factory. That is: one (and only one) partition 16MB large (sic) (it's a 4MB mem. stick, aka flash drive). As the typical size for FAT16 is 16MB, I make the file system FAT16 (that is, I assume it came with cluster size = eight 512-byte sectors). And this time, I succeed: the set DVD player - flash drive performs well.
However, I'll never know what the player was objecting: the corrupted file system or the lack of partitions/wrong file system. I now shall proceed by trial and error to make the full 4GB usable and I promise not to tell you anything about it. I'm sorry, but many a time I've been told, upon discovery of a solution, to state it in the forum. Greetings.