About one inch of the white wire from the end of the L shaped magsafe connector appears to be brown.
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It's odd that there are "burn" marks yet the cable doesn't get hot. The MagSafe cables have had an issue where flex at the top of the strain relief would eventually cause the insulation to crack and the conductors to short. The cable itself has a braid over the inner conductors and, with the insulation cracked, the conductors can short to the braid or to each other.
It may be that the cable isn't hot when you've touched it, but that the problem is intermittent. In fact, that's usually how cable failures start. Frankly there's probably no other explanation for the cable discoloration except heat. And the place yours is getting discolored is right at the top of the strain relief, which is where the cable is most likely to fail.
You might try holding the discolored section while you flex the cable around. You may feel it suddenly getting hot. The charge light will probably go out then as well. Keep in mind that, if this is the problem, the cable will eventually fail and that flexing it too much may make it fail sooner. If I were you I'd buy a spare sooner rather than later.
I had a MagSafe cable that failed at the top of the strain relief. Eventually the outer insulation sheath cracked completely. It wasn't long after that that the conductors began to short out from time to time. You could actually hear it and see the spark. Needless to say the insulation turned very dark very quickly. I replaced it before if caught fire.
The newer MagSafe adapters have much different cable than the earlier one I described above. The cable insulation seems much harder and less prone to breakdown. The trade-off being the newer cables are stiffer and a bit more difficult to coil up.
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One other note... I think officially the power adapter is considered a consumable item. It's a least arguable that the problem may have been caused by the user. (Yes I know, you've been very careful). Whether you can get it replaced under warranty will depend entirely on what you say and the mindset of the Apple rep you say it to. If you take it into an Apple store and tell the manager that you're afraid the thing is going to catch on fire, you'll be far more likely to get it replaced under warranty. Especially because they'll be able to look at the condition of the thing and evaluate if it was abused or not. You'll have less luck over the phone. In either case though, it won't be so much about whether it's in warranty as whether Apple wants to be liable for burning your house down, along with whatever kittens, puppies or babies may live there.
This is the line I'd use: "The twins are only four months old and, now that they're finally home from the hospital after their long, painful battle against that flesh-eating bacteria, I'm worried that my Apple power adapter is going to catch fire and I'll have to throw them from the eighth story window to keep them from being burned alive."
But then you might not be quite as warped an individual as I am.
