Hey all. I've been trying to figure out this quest given the confusing and overly wordy nature of the Second Outing on this wiki.
I found something on Reddit that gave a clear answer for those who just want to save him, regardless of who does it. Siegward or Patches.
Their way to do it was thus: Give Siegward his armor. Don't kill the Old Demon King. (Not necessary I believe but it was how he suggests) This will lock out Patches from saving him. Enter Irithyll of the Boreal Valley, reach the bonfire by the bridge, light it sit whatever. Just reach that point so Greirat will talk about it. Send him off. Reload the area so he vanishes. Kill the Old Demon King. Go back to Firelink and he should be back and he'll mention being saved by Siegward.
I just did it myself so I can confirm. The furthest into Irithyll I got was the church with Anri inside and I went no further. I didn't attempt to even go near the moat till I did the quest. I beat the ODK but I believe just any boss will do. I also beat the ODK with the Great Swamp summon NPC so there's no clash there.
Hope this helps.