Is this cheap extension cord safely rated for 1625W?
The difference between the two lies in isolation, not the conductor. Notice how the orange one is for outdoor usage, so it has to have one more layer of isolation.

For the 13 A current rating, AWG 14 would definitively suffice; that's a conductor diameter of 1.6 mm, so really not that much.

Also, the orange one has one conductor more – so, another reason it's thicker. Having seen industrial and event/stage cabling, I'd really not call your orange cable "heavy duty", by the way. It does look like the minimum effort you'd have to go through to rate something for outdoor usage.

So, I don't know why you'd distrust the rating of the white one; it looks fine for indoor usage, even if I don't like this slightly US-typical type of 2-conductor cabling, since it has a tendency to tangle.