The things you're describing aren't really version control systems themselves. Git is a version control system; these are an ecosystem of web-based tools surrounding that version control system.

I don't know if there's a good term for these.

kagis

Wikipedia calls them "forges":

https://en.wikipedia.org/wiki/Forge_(software)

In free and open-source software (FOSS) development communities, a forge is a web-based collaborative software platform for both developing and sharing computer applications.

For software developers it is an online service to host the tools they need to work and communicate with their coworkers. It provides a workflow to propose modifications and engage in discussions. The goal is to reach an agreement that will allow these modifications to be merged into the software repository.

For users, a forge is a repository of computer applications, a place where bugs can be reported, a channel to be informed of security issues, etc.

The source code itself is stored in a revision control system and linked to a wide range of services such as a code review, bug database, continuous integration, etc. When a development community forks, it duplicates the content of the forge and is then able to modify it without asking permission. A community may rely on services scattered on multiple forges: they are not necessarily hosted under the same domain.

Gogs is the original. Gitea is a fork because the dev of Gogs wasn't taking community input (I think that was the reasoning behind it). Forgejo is a fork of gitea because some folks didn't like gitea forming a for profit corporation (Or something to that effect).

As far as day to day use they're all fairly similar, though it's been a long time since I used Gogs.

You're missing GitLab. I'd be looking at GitLab or Forgejo.

But you might not need this. When you access a private Git repository, you're normally connecting over SSH and authenticating using SSH keys. By default, if you have Git installed on a server you can SSH to and you have a Git repository on that server in a location you can access, you can use that server as a Git remote. You only really want one these services if you want the CI pipelines or collaboration tools.

My only input is gitlab is very complicated, never used the others