Update Firmware of Samsung 840 Pro

Samsung is really, really weird and it took me many hours to figure this one out because it's absolutely counterintuitive.

It turned out that I was right with my skepticism of an image provided by Samsung probably actually being suitable to boot from it. Putting the image they offer you on a thumb drive doesn't work. It's not that it's super fast and you don't notice the update happening like I first thought, it's just that that image isn't bootable which means that nothing happens.

You have to mount that image, find a different image in it, and put that image onto your thumb drive. Because reasons, I guess.

Step-by-Step Guide

1. Check which firmware your SSD currently has via # hdparm -I /dev/sda. In my case it was Firmware Revision: DXM05B0Q.
2. Visit this site and under "Firmware" → "Samsung SSD Firmware for Windows Users" download "840 PRO Firmware" which currently has the description "ISO DXM06B0Q".
3. Mount the ISO file you just downloaded.
4. From the mountpoint, copy isolinux/btdsk.img to a different location. I'll assume /tmp/btdsk.img for it. This step is actually necessary because root can't read that file but your normal user account can.
5. Run sudo dd if=/tmp/btdsk.img of=/dev/sdb where /dev/sdb is your thumb drive. Go check whether it's /dev/sdb and make sure it's not mounted before you run the command! You will obviously lose the data stored on your thumb drive with this.
6. Shut your computer down.
7. Boot from the thumb drive.
8. It takes a few seconds, then you'll see the slightly confusing message "Firmware is already updated onto this SSD!". The firmware version is printed above it.
9. Press a key to continue.
10. You're shown some kind of shell. I didn't figure out how to reboot the computer from there so I simply killed it via a hard reset. So if you can't figure it out either, just press the power button for 6 seconds. If you figured it out, leave a comment or edit this answer.
11. Remove the thumb drive.
12. Boot your OS.
13. Run # hdparm -I /dev/sda, again, to verify the firmware has been updated. At the time of this writing, it says Firmware Revision: DXM06B0Q.
14. Burn your thumb drive to get rid of that software. Alternatively, delete its contents and reuse it.

Newer Firmware Versions

ens mentioned in the comments that newer firmware images can directly be copied to the thumb drive via dd without prior extraction from a different image. I have not tested this.

You need version 1.0 instead of 2.0 of the (console) Samsung Magician. Download links for it you can find at https://aur.archlinux.org/packages/samsung_magician-consumer-ssd/ (the Dropbox links).

See the pinned comment at https://aur.archlinux.org/packages/samsung_magician-consumer-ssd/:

Samsung has stopped providing a Linux version of the 'Magician' for Consumer SSDs. The newer v2.0 only works with Enterprise PM863 and SM863 Series. So this is the older version v1.0. It should still be working with most Consumer SSDs but be aware that this is not the latest updated version of this software!

BTW, the notes at https://wiki.archlinux.org/index.php/Solid_State_Drives#Samsung might also be helpful with your endeavor. Good luck! (Back when I tried to do the same for my EVO drive I created myself a Windows OTG flash drive using Rufus.)