Tbf though, Death is definitely a conscious being. Death doesn’t really decide who to kill till they die by natural means and grabs their Souls. Like you said he doesn’t need to physically do it. Death wanted to kill him, but changed his mind because he consciously decided to stop fighting him after Puss speech about knowing his fate will be him dying. The play with my food is mainly because he had been chasing Puss all this time, just to figure out Puss wasn’t the arrogant legend he was.
Difference between denying fate abstract and convincing a physical manifestation not to kill you.
Very much think of it like this, a conscious abstract that is able to form himself into a wolf.
It didn't feel like he 'changed' his mind, and Death would most certainly have rules, but Puss set the complete precedent and managed to escape him in this encounter despite literal Death coming for Puss. The reason Death spared Puss in that moment was because Puss was able to defy his previous self to the level that it changed when his final life was meant to die.
Death did this in a way by staring directly at Puss, or staring in the face of death, which is definitely more likely Death peering into Puss' actual soul than just merely looking at him given the nature of his being. In that way, Puss was able to bend the rules, 'ruin it' for Death as he put himself, and escape that whom no one else since Death existed could do. It certainly sounds like Puss defying his fate, which is the whole theme of the movie in itself. In the end, he got to live his last life cause he was finally able to understand what life is and cherish it.
Playing with his food more suggests Death was purposefully holding himself back to let Puss run. He does this, as its implied, because he hated Puss' arrogance, and multiple lives to go with it already being a defiance to Death, so he was likely trying to savour and enjoy Puss' fear, but left it too late in which Puss could change himself to a level that the concept of Death decided to let him live, but in a way that didn't seem like he had a choice in the matter.
The physical manifestation is still driven behind the concept of Death. He is literal death, and though you'd think he wouldnt be able to be bested somewhat like he was against Puss, its still defying Death. Puss remarks he 'cant beat him', so Puss will eventually die one day regardless of how much he's changed, but the fact he was able to push his pre-determined fate and defy Death just feels like he has some light passive Fate Manipulation.
I can grasp on what Death/Lobo is, and he does have a sense of conscience given his personality, clear biases, and the fact he was enjoying toying with Puss, rather than just being this stoic abstract concept that doesnt need to physically exist. But physical manifestation or not, it doesnt stop him from being Death, and it doesnt weaken the concept in itself. The fact Puss was able to keep living despite Death having his sights set on him from the start of the movie, because he managed to change and resist, is something a little more than Death, who clearly didnt want to let Puss keep living, but had no choice in the matter.