When the Romans have Jupiter, they're not going to call him Zeus; that's the Greeks' job. Likewise, the Japanese refer to that deity as "Enma", they don't need to acknowledge the "Yama" name outside of how they write it. The Japanese uses the kanji "閻羅刀" and the first two characters are the Han for "Yánluó", Yama the judge of the dead, so it's safe to say they're not referring to something like "Enma" that's written with the characters for "Monkey".
The connection is even easier to make comparing the JPN and ENG versions of Yu Yu Hakusho and Dragonball, with King Enma (閻羅大王,
Enma-daiou) in both anime being referred to King Yama in English sub or dub while the
text or
spoken Japanese is "Enma-daiou" with the same characters, and both are managers of the afterlife and spirits.
The idiocy of the Wiki is in acknowledging that the Yamato is written as "Enma Katana" (閻魔刀
Enma-tou) and read as "Yamatō" then making any mention whatsoever to 大和 "Yamato". Imagine selectively acknowledging a macron and then calling the words the same when they're not? A macron makes a ton of difference to legibility, it's why we pronounce the E differently in "extreme" than we would for "elephant". But I digress. It's the sword of the god of death that originally "[had] the power to divide and wipe out the darkness". Now it's who knows what doing god knows what, in the hands of some poetry nerd.