Yeah, a high res picture would help.
Two weeks?!!!!!!!!!! My cord doesn't get hot at the plug. The big white box gets hot, but that section of cord and plug doesn't even get warm to the touch. I don't think it's natural.
As far as burn through, as you said this one doesn't show it. While the other one showed it, I'd be interested in seeing if it also had pointy tooth-like marks.
You know, animals *can* be funny. If it was me and I bought a new cord, I'd be tempted to rub a little habanero sauce (Tabasco brand) on the wire. Your kitty might not like it, but then she might like it even less if she were to bite down to copper on a live line.
I look forward to a high res image later in the day. (My day is just about done.) Given what I saw in the low res pic and what you said, my guess is that your kitty is playing with it. I don't know why, maybe she likes the light when it's plugged into your machine, maybe there's something about it ithat's irresistable to her, but I think she could be gnawing on it and might have gotten zapped on the first cord.
Okay now I've seen the high res pic.
The wire is broken. The vinyl looks split. Cut? I don't know, but the copper wire is certainly cut. It wouldn't melt and look like that, especially with the vinal still looking like that. What you might see as melted vinyl is probably just stretched and crushed vinyl. Everything else seems fine. There's nothing drippy looking to it, so personally I'd rule out melting.
How does the back side look? Does it look fine, or does it show similar damage? If similar, then it could be your cat (though this new image doesn't look as much as if it was done by an animal), or it could be a wire cutter.
If the backside is fine, then my sense would be that it looks like the work of a little chisel (or screwdriver), or something sharp that might have dropped on it.
I don't know. It's really hard second guessing from a photo. I hope I've given you some ideas on how to look at it.
Hope this helps!