The song can be played anywhere on the piano and even Leonard Cohen might have changed that up over the years. You could start in C#4 if you want or F. I don't know what Leonard Cohen did on his first recording (whatever suited his voice and the vibe he wanted), but it doesn't have to matter much.
If the video and sheet music were not published together, it makes sense that they may vary.
For keys, it is generally easy for beginners to play in the keys of C or Am because you can more often ignore black keys that way.
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