This problem was such a headache for me. I am posting the solution here in case anyone else has this issue.
Problem: After trying to fix a problem in the game by the suggested method of unlinking then relinking my Steam account I could no longer launch the game. If I launched through Steam I would get an "Update Required" message that would close the launcher. If I opened the launcher directly through the game files then I could log in with the email address on my Steam account but I would get a message saying I needed to purchase or redeem the game. When I used the same email address to log into the Frontier website, I could see that my Steam account's email address (which is where I purchased the game) was linked to my Frontier account on that email address. So I could not identify where the problem was.
Solution: I had forgotten (mainly because I could never have expected this to be relevant) that years ago I had changed my primary email address on Steam. This was years after purchasing the game. Frontier support pointed out that the Elite Dangerous purchase key was associated with my old email address. So I had to open the launcher through the game files and log in with my old email address. After that, the game worked even when I launch from Steam.
One more thing to note: If you (like me) have changed your email address on Steam, then any purchases you make through Steam (Arx points and I assume DLC) will be associated with the new email address rather than the original one associated with your Elite Dangerous account. I am still waiting to hear back from support about how to get purchases applied to the ED account. I will update this post when I hear from them.
I hope this helps. I scoured the internet for solutions to this problem. It seems like there are a lot of things that can cause this "Update/Redeem" bug. This is a relatively easy troubleshoot so it might be worth a shot. If you do not have access to the original email address then I don't think there is any solution except to email Frontier support and ask them to transfer the purchase key.
Good luck!
