I think there's a slight confusion here, possibly caused by the use of language on-screen and in Atlassian's docs (not by us end-users!)
There are two things being discussed as though they are the same thing, and they are not.
As shown in the screenshot in the original question, these project types are a way of grouping projects together. They are supplied by the three applications - JIRA Core gives us "business", JIRA Software gives us "Software" and JIRA Service Desk would add "Service Desk" to the list.
Applications add these groups. I think they can also be added by add-ons, not just the applications.
These are different configurations used when creating a project, which try to help you get the config for a different style of project right. There are several templates for each project type. The templates also set the project type on the project when they are used to create a new project. They are all handled by (built in) add-ons, and you can write your own templates, and use the rather excellent Wittified add-on that Bart refers to for managing and creating them too.
So, either way, you're going to need to do something in an add-on to create a fourth type and it's not as simple as just creating a new project template.
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