Perceived power could also mean that while holding the sword the person has a different appearance? Perhaps of the last person that touched it? Or whose blood had previously stained the blade... Or...make you invisible! An invisible warrior! Sorry, I'm having fun here ^^
You could have it as perception of time as in it slows people around him/he speeds up, his perception of reality is enforced upon the world through the sword as a domain, if you've read Swallowed Star (excellent webfic) something like the origin laws channeled through the sword, really perception is such an abstract concept you could say what he thinks it does becomes its powers or it channels his imagination into reality or it creates illusions and takes power from defeated opponents to enforce them onto reality, you could say it bends space or light in every dimension and he can move freely in this area while everyone else is effectively blinded, or teleporting randomly as they move. The last one could lead to an interesting fight between the wielder and someone who understands space more than hims and walks circles around him in a Hall of Mirrors sort of way.
Perceived power could also mean it changes the person more into who they wish they were or think they are but they can't control it. This would be more of a side thing than the main power.
You could have the idea it helps the wielder perceive reality more like notice little things and plots by the enemy or it produces an aura that makes others naturally see him as what he wishes to be seen as. For example he it fighting a battle it gives off the aura of a powerful warrior or in a debate as a powerful and knowing speaker. Except there is no off button and everyone starts to always see him as the perfect hero or all-knowing almost setting him up for failure and the reaction to failure is judged by the sword as to whether he is capable of ignoring other's opinions and knowing themselves for what they actually are as a test as to whether he is worthy of the sword's true power's.
