While we certainly may never know how exactly these came about in terms of being added to the lists, let's still take a general look at these "oddities", starting with what (technically) works on the Xbox 360.
Support for OG Xbox games on 360 was done via an emulator named Fusion that was developed in-house at Microsoft, and because of just how different the two consoles were it was even deemed "impossible" by half of the Xbox team; even those who worked on Fusion called it "magical". While all 996 OG Xbox games are supported by Fusion in theory, Microsoft limited official support to just 463 games, or ~46.49% of the entire catalog, due to a wide variety of games having some sort of technical issue (& even many supported games don't exactly play 100% like they should). Two years after the console launched, the Xbox Originals program debuted, which digitally re-released certain OG Xbox games that were already BC, and it turns out that the two Japan-exclusive BC titles were also re-released as Xbox Originals! First up is 2003's Magatama, which is arguably the most interesting Japan-exclusive OG Xbox game, as it's not only one of only five games published by Microsoft itself that wound up being exclusive to the region but also the first one that was actually developed in-house at Xbox Game Studios Japan, the same studio (though with a different team) that would later make cult-classic Phantom Dust. Specifically, it was produced by Hiroshi Kawai, an ex-Sqauresoft employee who was actually the main programmer behind Final Fantasy VII & IX, two of the most beloved entries in the entire franchise; the actual director was Daisuke Fukugawa, who would later direct cult-classic RPG Lost Odyssey. A third-person action title, Magatama is today in Japan considered both a "Fukishin/Unscrupulous Game", due to it directly addressing a major real-world event (in this case the Warring States period), & an example of "Kiku/Chrysanthemum Taboo", as it criticizes or parodies the Japanese Imperial Family, which is generally considered off-limits; the Imperial Family crest is a Chrysanthemum, hence the name. Likely due to its immensely Japan-centric focus, & a lack of interest from any other publishers, Magatama never saw release outside of Japan, and due to the OG Xbox's region lock it's not easily playable today, outside of getting a Japanese Xbox or soft-modding a console. Sadly, said region lock also applies to the Xbox 360, so you can't simply buy a physical copy, put it in just any 360 console, even change the "Region" to Japan, & start playing; it has to be a Japanese 360.
If you want more detail about Magatama, Import Gaming FTW! covered it well back in 2014. With both Phantom Dust & Metal Wolf Chaos having received great HD remasters on Xbox One, maybe we should finally get one for Magatama on the Xbox Series, so that it can finally see release outside of Japan.