Magic Banana suggested:
>> There apparently is something wrong written at line 11 or 12 of /etc/grub.d/01_PASSWORD ...

Here's what I gan gather from the various HDD's when poking around:

>>> 1TB USB-HDD:
sudo cat 01_PASSWORD:
...
10: echo set superusers=grub
11: # echo password grub 31475
12: echo password grub 5807

sudo update-grub2
Generating grub configuration file ...
Found linux image: /boot/vmlinuz-3.13.0-79-lowlatency
Found initrd image: /boot/initrd.img-3.13.0-79-lowlatency
...
Found linux image: /boot/vmlinuz-3.13.0-68-lowlatency
Found initrd image: /boot/initrd.img-3.13.0-68-lowlatency
  No volume groups found
Found Trisquel GNU/Linux 7.0, Belenos (7.0) on /dev/sda5
done

>>> 320GB Int-HDD:
sudo cat 01_PASSWORD:
...
10: echo set superusers=grub
11: * echo password grub 2127
12: echo password grub 5807

sudo update-grub2
Generating grub configuration file ...
/etc/grub.d/01_PASSWORD: 11: /etc/grub.d/01_PASSWORD: 00_header: not found

>>> 500GB USB-HDD:
sudo cat 00_PASSWORD:
...
10: echo set superusers=grub
11: * echo password grub 26753
12: echo password grub 5807

sudo update-grub2
Generating grub configuration file ...
/etc/grub.d/01_PASSWORD: 11: /etc/grub.d/01_PASSWORD: 00_header: not found


>>> 40GB USB-HDD:
sudo cat 00_PASSWORD:
...
10: echo set superusers=grub
11: * echo password grub 9410
12: echo password grub 5807

>>> Immediately after editing 01_PASSWORD I got:
sudo update-grub2
Generating grub configuration file ...
/etc/grub.d/01_PASSWORD: 12: /etc/grub.d/01_PASSWORD: 00_header: not found

>>> Subsequently, without editing 01_PASSWORD I get:
sudo update-grub2
Generating grub configuration file ...
/etc/grub.d/01_PASSWORD: 11: /etc/grub.d/01_PASSWORD: 00_header: not found

>>> Now, re-running Magic Banana's suggestion:
sudo dpkg-reconfigure grub-pc
Installing for i386-pc platform.
Installation finished. No error reported.
Generating grub configuration file ...
/etc/grub.d/01_PASSWORD: 11: /etc/grub.d/01_PASSWORD: 00_header: not found

Note: there isn't any /boot/grub/menu.lst file, either in the USB-HDD or in the internal HDD (whose /boot folder isn't in the first partition; that's occupied by the original Windows_7 OS). It turns out that I can't find any example of that /boot/grub/menu.lst file in the 500GB-USB-HDD, in the 40GB-USB-HDD or even in the 1TB-USB-HDD (where update-grub2 works OK).

I looked in two of the /etc/grub.d/00_header files (Int-320GB-HDD and 1TB-USB-HDD) ... no mention of PASSWORD found.

My next step will be to do the suggested "comment-out" step on a couple of these HDD's.

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