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I remember, that's a great example.So, basically;
When Kratos opens Pandora's Box the first time, he gains the Power of Hope and uses a small measure of it to kill Ares. But after that, it gets locked away again under his despair and he shortly gets Ares' mantle and power as the god of war.
When he attacks Rhodes, Zeus tricks him into giving up his godhood and kills him. After which he is brought back and goes on a journey to the Sisters of Fate to turn back time. When he does this, his base power grows to the point of being on par with his godly self by the time he reaches the Loom.
So at that point, his innate power (boosted by the Titans' magic) is godly. He regains his godly powers and immortality from the Blade of Olympus but not the Godhead (the specific mantle of war).
When he sacrificed himself to release Hope, he seemingly gave up all of his power but when he came back to life, his innate strength became on par with his peak godly self, when he was bolstered by his Titan powers and godly magics, as well as Hades' soul.
In essence, it's kinda like the SSG/base form situation with Goku.
He's technically an immortal demigod but as far as the Norse realms are concerned, he's a god considering his immortality and power. Their definition of godhood is very nebulous given Atreus calls himself a god, Thor calls himself a god etc. despite technically being Demigods. Even Aesir/Vanir are just races and not innately gods.