I am not sure exactly where the confusion lies, the names and description of the different integration points I feel are spot on.
Web App is basically for any middleware system that you set in place. As this is able to be accessed and processed via 'Client-side' (e.g. Web browser), this is not the most secure place to store your id and secret that you use to create the token. So to help combat this inherit risk, they have added a few more steps in to ensure it is you that is authenticating and not someone that is replicating your process.
Public App is best described as an actual application (whether on phone, computer, tablet, etc) that is stored locally on a users device. This is very high risk, so it requires the most stringent security methods to be able to authenticate against your SFMC instance.
Server-to-Server is exactly as it states, your server directly connecting to SFMC server. If you plan on having an outside CRM or analytics tool access or manipulate data in SFMC, this is likely to be the option you choose.