The method by which you make soap (hot process or cold) doesn't matter after the soap is made and cured. What can make a difference in the finished soap is the proportions and types of the fats in the recipe, the amount of superfat, and other additives.
I'm seeing a different ingredient list on their website -- sodium palmate, sodium palm kernelate, water, glycerin, sorbitol, sodium gluconate, palm acid, sodium chloride, palm kernel acid. Image source:
https://cdn.shopify.com/s/files/1/0266/7171/0307/products/0020_baleco30aout201933133_1800x1800.jpg
To be honest, this reads as much like an everyday basic soap as it does a "glycerin" soap. The ingredients list is written using the "what comes out of the soap pot" method.
Glycerin in an ingredients list like this can easily come from the process of soap making; it doesn't have to be an added ingredient. Most of the other ingredients are also normal chemicals created during the soap making process. The only exceptions would be sodium gluconate (a chelator that helps preserve the soap), sorbitol (a sugar alcohol that adds lather), and sodium chloride (table salt, adds hardness).
Maybe Carolyn
@cmzaha can give some insight about the sorbitol -- she uses it in her soap making.