I tried quite a lot of things to get this achievement, read all the guides and all discussions regarding this achievement. Finally, I found a way to get this. Thanks goes to marklar and this discussion.
To have a chance to achieve "Bad Luck Cultist Extravaganza", you must downgrade the game to the state of the 9th February 2020 (or wait for the Prologue Patch the developers promissed in this discussion in July 2021).

1. Open the Steam console (Copy this ==> steam://nav/console <== into your browser, open with Steam Client Bootstraper).
   
   
2. Type in the following command
   download_depot 869760 869761 9059420696505735575
   
   
3. Wait for the download to finish. You will not see any progress until finished.
   
   
4. Open the folder where the current Stoneshard: Prologue is saved (something like /YOUR_STEAM_LOCATION/steamapps/common/Stoneshard Prologue/). Copy the content of this folder somewhere save so you have a backup.
   
   
5. Remove the content of this folder.
   
   
6. Open the folder where the older version was downloaded (something like /YOUR_STEAM_LOCATION/steamapps/content/app_869760/depot_869761/).
   
   
7. Copy the content of the older version folder to the emptied Stoneshard Prologue-folder.
   
   
8. Rename "StoneShard.exe" into "Stoneshard Prologue.exe", otherwise the game will not start.
   
   
9. Start the game via the "Stoneshard Prologue.exe"-file. Steam will update your game files (thats a little bit annoying).
   
   
10. Let the Stoneshard: Prologue update stay in your "active downloads".
    
    
11. Empty the real game folder and copy the old version into this game folder again.
    
    
12. Rename the exe-file again.
    
    
13. Now your game should start without updating.

If you're playing the older version now, you should see some little differences:

• The lockpick-option at doors and chests should be named differently (at least in the german version).
  
  
• The first enemy should see you a little bit later.
  
  
• The scrolls should have different sprites.
  
  
• The two dungeon levels you are playing should be a little bit larger and you find more items.

If none of above things are in your game, it's most likely that the downgrade failed. Try starting again.

Now the part, I don't understand myself yet. I don't know, which enemies you must kill to get the achievement. These are not the ones mentioned in this guide.

Here is, what I did:

• Started the game, gone throug the initial level down to the scrolls room.
  
  
• Killed quite a lot different cultists (at least 5 different), some rats and bats.
  
  
• Died and loaded the game (starting in the scrolls room).
  
  
• Killed only some cultists (also encountered in the first run), bats and rats.
  
  
• Died again (I never even tried at this point, lol) and loaded the game (starting in the scrolls room).
  
  
• Killed only 3 different cultists (the one that seems damaged, the crawling one and one adept) and the achievement popped after the last one. I think "one run" counts as "as long as you're not in the main menu" and because I loaded the game everytime, the "6 different cultists" to kill counted over all 3 runs. The adept was the one I didn't encountered in the two runs before.

Note: Everytime you die and load the game (starting at the scrolls room) the following 2 dungeon levels are generated randomly. Therefore you can encounter different cultists.

I hope this guide helps some people to get this achievement without digging so deep into this like I did.

Some helpful links:

Game downgrade guide

SteamDB to find older builds for Stoneshard: Prologue[steamdb.info]

A helpful guide for the other achievements

A discussion abount Bad Luck Vampire Extravaganza (the achievement before Bad Luck Cultist Extravaganza)

The devs promise to upgrade the prologue to patch this achievement