What are the most known start-ups business models other than a marketplace, SaaS, an on-demand service, and e-commerce model?

First of all I want to clarify one thing on your question that I believe it's not accurate: SaaS is a way to deliver technology (products) not a business model. The counter part of SaaS is on-premise or legacy, which is not a business model but how you deliver your product.
For example, Salesforce is SaaS and their business model is charging per seat or user. Zenefits is a SaaS model and they don't charge the user of the product but take a commission out of insurances. Our company, COMPAS, is SaaS and we charge a monthly fee on an annual contract.

Sorry I didn't answer your question directly but I believe it's more important to ask the right questions before you get the wrong answers.

The question you are asking appears not to be business models but types of business you apply to a business model.

I found this great article on Huffington Post which gives a greater understanding of what business models are and why they work.

http://www.huffingtonpost.com/nina-tomaro/9-proven-business-models-_b_7949932.html

The article lists 9 which are commonly used:

* The Middle Man
* Become a Marketplace
* The Subscription Model
* Customize Everything
* On-Demand
* The Modernized Direct Sales Model
* Freemium Model
* Reverse Auction
* Virtual Good Model

I would also Like to add to that list

* Professional Services - Project Management, Software Development, Networking, Systems Administration, Business consulting, Management Consulting and other Consulting work. (Examples Robert Half consulting, TeKSystems, Aerotek ...)
* Value Added Reseller (VAR) - Similar to Customize Everything, except you resell and add cusomizations or provide other services bundled with the items sold.

There are many business models and variation on the above business models. Many businesses have hybrid business models combining aspects from several models. Some of them are well implemented and successful but not all achieve great success.