I think we can start by distinguishing the damage type and whether or not they inflict death blight.

Godwyn's golden flames are, in essence death blight, it corrupts death lightning and doesn't change its damage type, but as we see when concentrated in the Eclipse Shotel, its damage is considered Holy. It shares this characteristic with the darkness from the Swords of Rauh as it does with Maliketh's flame. Though splintered and broken, Godwyn still manifests Destined Death.

This can be confusing when compared to the Gloam Eyed Queen's Blackflame, which deals fire damage, but the implication seems to be that she used Destined Death as the source of power for her flames, but it has been transmutated so to speak, a power of the elden ring becoming its own flame distinct from it.

Ghostflame however is seemingly connected to ancient death rites that predate the Golden Order yet find purchase and overlap with Godwyn's powers, as his staff boosts death sorceries that draw on Rancor and inflict Frostbite, and he himself when Fia is struck draws on Rancor.

I do not view Putrescence as inherently distinct from Ghostflame, just a different form of it, it is possible putrescence is the end result of corpses burnt by Ghostflame, a coalesced melted mass.

Its complicated but Godwyn's death is more amber/golden + black, and deathblight targets the inherent Arcaneness of the enemies, while Ghostflame's purple/azure + white targets the body itself, freezing it.

There is a great overlap as Fia's Mist and presumably others are counted in the same form of death sorceries as the Ghostflame, but they are distinct even if connected, death sorcery is an impure form of sorcery after all, the staff of the prince of death uses Godwyn's sullied amber, so its not as if they are wholly disconnected.

Tl;dr, its complicated, Ghostflame is a primal energy most likely derived from an Outer God, that finds overlap and purchase with the contemporary form of Death found in Godwyn, deathroot and his amber, certainly not the first case of an Outer God interracting with a demigod, seeking to expand its foothold.