It is John but not a 1965 demo. John Robertson in 'The Art and Music of John Lennon' has a Help! run-thru demo listed as being recorded by John in December 1970 along with 'I'm The Greatest', 'Make Love Not War', 'How', 'Oh Yoko' and 'Sally and Billy' and writes
"During his interview with Jann Wenner for Rolling Stone, Lennon expressed the desire to re-record Help! at the slower tempo in which he'd originally written the song, rather than the hurried take preferred as The Beatles single release. He devoted a minute or so during this composing tape to try and perfect a new arrangement, ...