This description is misleading! The trophy will only unlock If you beat the boss on the US and EU vers. Doesn't matter wich one, for me the first boss is very unfair and is really hard to beat If you die and come back with empty hands from the last check point. The biggest challenge on that boss besides the barriers get fast is the totally unwanted "inverted" item that come from nowhere and mess up the battle, easy to die mainly when you are dodging the barriers.
My solution was, upgrade until Lv.4 and start the second stage. The entire stage is way more easy to finish and even starting from the last checkpoint is doable beat the second boss.
On the Japanese (JP) and Asia (AS) vers., this trophy will unlock when you encounter the boss, no fight is necessary. For a unkown reason in the other vers. beat the boss is a mandatory.