No matter which editing software you are using, it is a bad idea to overwrite the original file with your changes. You should always keep the original around because you might want to make different changes at a later point, e.g. because you made a mistake or because you have learned a new editing technique.

So your first approach was right: you should export your files from darktable, which will result in you having the original file and the edited one separately. You can export them into the same directory or a different one, I suspect most people do the latter and have a directory just for edited files. If disk space is such a major problem for you, obviously you can delete one of the files when you think no longer need it, but disk space is actually fairly cheap nowadays.

Your second approach is definitely not how you are meant to use darktable: the xmp file describes the edits to the original image, you overwrote that with the edited one, so now it is applying the edits described by the xmp file to the already edited file. The result is wrong for obvious reasons.

darktable, is, according to its own FAQ, not a file manager. (I actually remembered a differently worded version of that.) I can't answer about "like with Lightroom" because I have never used Lightroom; you shouldn't assume everyone has.