I changed also several Suunto Eon, Vyper and Cobra batteries and never changed the o-ring as
i could not find the original repair kit from Suunto, BUT don't do this, get the Suunto kit to be safer.
Anyway, as LiquidGuru already said, thake the straps off to have full, free access to the back and
it needs a bit of practice to hold pressed down the black ring while inserting the small black piece (retaining band?)
and turn to lock.
The important thing is to double check that the transparent cap is not pinching the o-ring while keep it pressed
and then put the ring and the retaining band on without releasing the pressure or rotate the cap.
The best way to do it - and different as shown at 1.20 in the video - is to slide the ring over your finger,
press then the transparent cap down and check carefully that the o-ring sit in its place and then just let the ring slip
down your finger in order to NOT release the pressure on the cap and twist the ring.
If you have a divider around use it to twist the ring instead of pushing only on one hole with a screw driver or similar,
this will help to turn the ring even and avoid that he don't lock well in place.
Chris