Difference between Soap and Detergent
There is no restriction that soap be natural or unnatural. Soap are salts and detergents are not necessarily salts. The latter is a very broad term. Soap has a narrow meaning.
IUPAC clearly defines soap as

&quot;A salt of a fatty acid, saturated or unsaturated, containing at least
eight carbon atoms or a mixture of such salts.&quot;

Just note that not all soaps are used for cleaning nor there is any condition that they should be soluble in water. Insoluble soaps of metallic salts serve as lubricants.
Similarly,
IUPAC defines a detergent as

&quot;A surfactant (or a mixture containing one or more surfactants) having
cleaning properties in dilute solutions (soaps are surfactants and
detergents)&quot;

Just like banana is a fruit but not all fruits are bananas, in the same way, all water soluble soaps are detergents and surfactants. Not all detergents are soaps.
Visit the IUPAC Gold Book and search the entries detergents and soap.