do a search on the chemistry of soap for a more detailed explanation, but essentially, soap is the result of reacting fatty acids (oils) with caustic (lye). it does result in some glycerin, but also in sodium (if using NaOH) or potassium (if using KOH) salts of the fatty acids - also known as SOAP.
true soap is made using a caustic (lye). some soapmakers choose not to show this on their labels. our soaps are a mix of water, salts of fatty acids, excess oils, and glycerin, plus whatever else we put in there.
commercial soapmakers (like P&G) make soaps very differently. if they are making real soap (the salts of fatty acids), they typically purchase (or make elsewhere) these salts of fatty acids and they purify them - so they don't have the mix of ingredients unless they choose to put them in there. and they don't typically list the caustic. many times they are not selling "soap" per se, but actually detergent products which - chemically speaking - are another animal altogether. they act the same way soap does - as emulsifiers - but are different. detergents are not soaps and are not produced by mixing lye with oils.
oh, and many commercial soaps are a blend of soap & detergent.
yes - adding lye results in glycerin, AND SOAP. glycerin is not the same as soap.
if you don't want to handle lye, you can purchase pre-made soap or a detergent base. it's called melt & pour and is fun to work with.