Tallow, lard, palm and other fats can be separated fairly easily into "stearin" and "olein" by cooling the melted fat slowly until some it solidifies. The solid part of the fat is filtered out and called stearin. The liquid part of the fat that remains is the olein. Like any other triglyceride fat, stearin (or olein) creates glycerin when it is saponified.
Palm stearin is just stearin made from palm oil. It's the most common stearin that most people talk about here on SMF. You'd use it exactly like you would use regular palm oil. The stearic and palmitic acids in soap made with stearin might be a bit higher than in soap made with regular palm oil, but it's not going to be a huge difference.
You wrote:
"...I read somewhere that adding palm stearin will give the soap a higher percentage of glycerin..."
Stearin is about the same as any other soap making fat as far as the amount of glycerin created when it's saponified. You can calculate it:
https://classicbells.com/soap/glycerin.html
Stearic acid is made by further processing stearin to break the fat molecules apart into glycerin and fatty acid molecules. The glycerin is removed and the fatty acids that remain are either sold as commercial grade stearic acid, which is a mix of mostly stearic and palmitic acids, or further refined. Stearic acid adds no glycerin when it saponifies.
The glycerin from fats vs no glycerin from fatty acids could be the point the author was making in the source you read. Or the author might simply be parroting bad facts from another (misinformed) source without checking. Because you haven't provided a link to the source, it's hard to say.
Stearin will saponify slowly just like any normal soaping fat. Stearic acid saponifies very quickly, just like any fatty acid. It's stearic acid that you would want to use with care. Unless you're making shave soap or cream soap, most people don't generally use a lot of stearic acid in their recipes -- I think I've seen mentions of maybe 2-3% stearic acid for regular bathing soap.