I guess the word falsely depends on how The Joker has been portrayed. For a time, Batman's creators reluctantly changed their characters, including The Joker, to comply with the Comics Code Authority. The CCA's imposed censorship forced them to showcase mockeries of their characters for 'society's sake'. With the passage of time, they returned to representing their fictional personalities as they were originally intended.
If not for alterations mandated by the CCA, the Batman, Joker, etc., seen in comics and television during the 50's and 60's, in addition to their appearances in cartoon and animated serieses during the 70's, 80's, 90's and this decade, may never have been. The original versions of the characters from the 40's who were re-adopted in the mid-70's forward in comics would have been the only representations for the public to love, hate or be indifferent to.
Heath Ledgers rendition of The Joker remains true to the version created BY its creator. All other versions, no matter how popular, are parodies, lampoons and spoofs of the Bob Kane's masterwork. 'False' may be a strong word, but it is accurate in this case.