Despite linking my Twitch to my Microsoft (Xbox) Klei Account before watching a stream with drops enabled, I was not able to claim the reward on Xbox. The first reward (the icon) was claimable on my PC version through Steam, though, from when I linked my Steam and Twitch years ago.
Thinking having two accounts linked may be the issue (I want the rewards on Xbox, not Steam, since that's the platform I play on most often these days), I unlinked everything and then relinked my Microsoft Klei to my Twitch. The second reward tier rolled around... and while it says I claimed it and a game account is connected, I did not get the reward on either account.
All that's left now is the actual skin (although I haven't watched enough yet to unlock it). I want to ensure this error is fixed before reaching that point and attempting to claim the skin. ((If this is somehow a human error, and not a bug, I'd appreciate help in correcting my mistake!))
1) Link Klei Account to Twitch
2) Watch a DST stream on Twitch with drops enabled
3) Access Drops in your Twitch Inventory to Claim Reward