I personally can't stand flatpaks, they're too large to update its libs every few days. On a 4 MB/sec line that I'm, it's not a good option.

what’s called Logithèque in French, and maybe Software manager (or “repository”) in English?

Sounds like the distribution's "package manager," or rather, one of the GUIs for it. (Not sure which one because I don't know what distro you're using).

In order to understand what things like "AppImage" and other types of "packages" are, I think it helps to know the history.

TL;DR:

Since GNU/Linux is a Free Software OS, in the beginning, everybody had to compile their software from source and manage the dependencies for it manually. This sucked, so Linux distributions started curating the software and "packaging" it to reference its dependencies (that's why they're called "distributions" in the first place, BTW). The package manager (e.g. Debian's dpkg/APT and Red Hat's RPM) can recursively check for and install all those correct dependencies before installing the program you actually asked it for.

But even that wasn't good enough (apparently) because it happened on a per-distro basis and because (generally speaking) all the software on the system shared the same version of each dependency, so all the software had to be kept updated in lockstep. So now people have come up with systems like Flatpak and AppImage that are designed to work across distros, include the dependencies with the apps themselves so they aren't shared with the rest of the system, and in some cases also run the program isolated in a container for better security.

Bottom line is, now there are several ways to install software, with various tradeoffs:

• Using your distro's package manager: easy one-command install, integrates with the system
• Using a universal package, like Flatpak, AppImage, or Snap: easy one-command install, keeps the program relatively isolated from the rest of the system
• Compiling from source, the old way: complicated, but occasionally necessary in very niche cases, such as when the program is really obscure or when you actually want to take advantage of the whole "Free Software" thing and modify it yourself.

One last thing: GUIs for this stuff started out as interfaces for the distro's package manager, but are increasingly designed to work with universal packages as well. With the command line it's generally obvious what kind of package you're installing (because you're typing a different command entirely), but with the GUI you have to pay close attention if you actually care whether you're installing a dpkg or a flatpak. Also, if you see an app listed twice, that's probably the reason why.