So sodium stearate is not a soap?
I was always under impression that detergent refers to the mode of action, and technically soaps are detergents.
Beware: seems like you are falling into a trap of "natural is safe, artificial is harmful". It doesn't work this way. Some things are safe, some are harmful, some are natural, some are artificial - these are completely unrelated classifications, and there is no correlation between them. Plenty of natural, dangerous things, plenty of artificial, perfectly safe ones. Each should be judged separately.
Solution of typical soaps - salts of higher carboxylic acids - are always basic (have high pH, because of hydrolysis). That never works well for the skin. Artificial detergents are not limited this way, solutions of some of the are neutral, or can work even when slightly acidic - and that's much better for the skin. It doesn't mean every artificial detergent is good, again - each should be judged separately.