The weapon you have (jitte/jutte) is 十手, "ten hands", which can be read as "power of ten hands weapon". In hiragana, じって, which has Hepburn romanisation "jitte". "Jitte" is the correct romanisation of the Japanese name of the weapon.
The usual romanisation of 十, "ten", is "ju", and the usual romanisation of 手, "hand", is "te" (as in "kara-te"), so we find "jutte", though this doesn't reflect the Japanese hiragana writing or the Japanese pronunciation. It could be argued that "jutte" is the correct (or incorrect) English name for the weapon.
The spear-point jitte shown above is, according to Serge Mol, "Classical Weaponry of Japan", 實手, or "real hand"/"true hand" (or "fruit hand"). I haven't seen this elsewhere, and as far as I can tell, 實 (or the equivalent 実) doesn't have a romanisation of "ji" or "ju". For romanisations of these, see:
https://en.wiktionary.org/wiki/%E5%AF%A6#Japanese
https://en.wiktionary.org/wiki/%E5%AE%9F#Japanese
Mol uses "jutte" for the truncheon to distinguish between the two weapons, akin to "kris" and "keris" being used to distinguish between Philippine and Indonesian versions.
Is either jutte or jitte given in the OED?
(Compare "jujutsu", "jujitsu", "jiujitsu" as variant romanisations of the same word.)