I assume that the risk of using various power saws does differ, based on the wide spectrum of power saw sizes and designs.
This is why your question has generated so much confusion: you're mistaken, and powerpower saws do not appreciably differ mostly in what jobs they are best at, not so much in their safety characteristics. They're not like cars, where one is a tank and another is a flaming deathtrap. Powersaws are generally very simple machines: a motor with a blade on it, in some configuration. Basic safety features like a blade guard are standard on all. Advancedmoderate safety features like an electric brake doesn't actually make a power saw all that much safer. The SawStop has an advanced safety feature that has been talked about before, but it's very expensive and it's a table saw, which is good for some kinds of cuts and less good for others.
With power toolssaws, safety comes from the operator, not the tool.
There is no power saw that I would be comfortable giving to an average 13 year-old. However, a 13 year-old who is an apprentice carpenter or construction worker, who has been around power tools for years already... this kid I would feel comfortable giving any power saw to.
All that said, let me echo the recommendation for a jigsaw. Jigsaws are probably harder to accidentally mangle yourself with, and can handle just about any cutting task. The downside is that they can cut slowly, so there may not be much of an advantage over your manual saws if speed is what you're after.