It's played in C position but is in the key of B if that makes sense to you. You tune the whole bass down a half step, so the bass is tuned to E flat and you play it in C position although it's key of B, lots of bands did that, Hendrix and Van Helan are two that pop out immediately, I think the guitarists liked it better as it made it easier to bend strings, especially before they had the skinny strings, I know for example Mike Bloomfield used to use a banjo string for his high E string for example.
Another thing I noticed is that in the finished version the bass has a lot less treble than that video with just bass you put in the other day, I would guess the track you put on the other day was just a rough bass track with no studio EQ applied. Try playing along with this, it's in B and the lowest note is E flat. I doubt any cover band would tune the whole band down a half step just to play this in the original key. Whoever played that bass was a very good player, the part really moves and gives the song energy. Those guys were great musicians anyway.
I happened to be working an overnight job the night their plane went down and was listening to the radio and heard the announcement at about 4- 4:30 am IIRC.