It's simple. You get a Proof of a Concord Well Kept if you get summoned as a Blade of the Darkmoon and a red dark spirit who has the Aldrich Faithful covenant equipped dies. Not someone invading as an Aldrich Faithful, but someone who used the Red Eye Orb to invade with the covenant on. You cannot be a Blue Sentinel.
Also, sin (i.e. aggroing friendly NPCs in your world) has absolutely no effect.
Me, u/Domperidone (aka Dom) and his friend /u/endoasumi (aka Buttercup) set out to test this by taking low level characters (12-20), upgrading their weapons to +7 and bringing them to Ariandel Chapel, thus ensuring that out of all the players we would be the only ones within matchmaking range of each other. I hosted with Way of Blue and Dried Finger on.
- Buttercup killed Dom as a Blade of the Darkmoon while he had no covenant equipped, and had sinned as determined by the Statue of Velka. Ten times out of ten Buttercup got a single Proof.
- Buttercup then killed Dom while he had the Aldrich Faithful covenant equipped, and had sinned. Five times out of five Buttercup got a Proof Well Kept.
- Dom then paid to remove his sin, and kept AF on. Four times out of four Buttercup got a Proof Well Kept.
- Buttercup equipped Blue Sentinels and killed Dom, who had AF on. Buttercup got a single Proof.
- Dom equipped Rosaria's Fingers and was killed by Buttercup as a BotDM. Buttercup got a single Proof.
- The same, but with Watchdogs of Farron. Buttercup got a single Proof.
- The same, but with Blue Sentinels and Watchdogs of Farron. Buttercup got a single Proof.
In summary, in all nine occasions of a BotDM killing an AF red invader, a Proof Well Kept was obtained. In all 12 occasions of a BotDM killing a red invader of a different (or no) covenant, a single Proof was obtained. In both occasions of a Blue Sentinel killing a red invader of any covenant, a single Proof was obtained.
This is consistent with the tests in this post, in which it was speculated that only BotDM could obtain Proofs Well Kept when killing red invaders with AF equipped. We did not rely on Blues summoned by chance; us three were the only participants, so we controlled all variables. This makes our tests very reliable.
Unfortunately the recording of the tests failed, all we have is some screencaps. But this is easy to replicate: all you need are three people with characters in limited matchmaking pools (high level with low upgrade levels, or low level with high upgrade levels). If those are difficult to obtain, simply make one such character and copy the savefile to two other people.
By the gods, if I knew it was going to be this easy to prove, I'd have done it a long time ago.