The AppImage is one file that needs no installing and no root.
Mark it executable and run it — on some distributions it also
wants FUSE 2, which is libfuse2 or libfuse2t64
in your package manager. The .deb is for Debian, Ubuntu and
anything built on them: sudo apt install ./<the file>,
and the app arrives in the menu like any other. Either can update
itself in place afterwards.
Linux will not warn you about any of this, which is worth saying out
loud rather than leaving as a pleasant surprise. Nothing here is signed
by a distribution, nobody has reviewed it into a repository, and the
checksum below is the whole of what stands in for that.