It's an interesting question, but open to interpretation, because 'type' is rather a broad term. I used the words 'types of soaps' to do a Google search and found lists with what I would consider completely unrelated categories within each of the lists I found.
For example, the HandCrafted Soap & Cosmetic Guild has this listing:
https://www.soapguild.org/consumers/types-of-soap.php
The Soap History website (not the only website dedicated to soap history, btw) has this listing:
http://www.soaphistory.net/soap-facts/soap-types/
If you compare the two listings, I'd say it's like comparing apples to oranges.
Greenyards list types of soap by two different criteria; the first the process by which they are made and the second by usage:
http://greenyards.com.my/2017/02/25/types-of-soaps/
So apparently there are lists out there, but not lists that seem to really be what you are looking for from your OP and it would seem your list is not solely based on usage, nor on process either.
To make a list of soaps that have been made, I think you best bet is to look perhaps at a site such as Millers to see hundreds of different recipes and soaps that have been made. The purpose of each soap may or may not be listed along with the recipe, but they sure do have a large variety of recipes.
http://www.millersoap.com/
Other than that, if I were on the search, I would first determine what my criteria was or meaning of 'types of soap' and after I defined that, then I'd go looking for listings using the most definitive words in my searches. Sometimes honing our search parameters really helps to find what we are looking for, and without a bit of that, the results are too broad and unsatisfying.
Another place to look might be in various soap making books and that might give you an idea of different types of soaps that can be made. There are many many many soap making books, and once you've read several and made several batches of soap, you may discover that they aren't all excellent, but that's another topic for another thread (there are threads here with soap making book recommendations, btw.)