In reply to a_faulding: This is one of the most insightful questions posted on here recently.
Figure-of-eights are vastly more complicated than most people (even most climbing instructors) realise. However, before I start an in-depth discourse it is worth emphasising that even extremely untidy Figure-of-eight knots are still perfectly safe and are NOT significantly weaker.
Its forgiving nature is one of the main reasons why we use it.
[See the paragraph 'Myths about backup knots and neatness of knots' at the bottom of the third page of
http://www.efclimbers.net/resources/Knot-and-cord-strength.pdf if you want some evidence about this.]
Even a basic single strand Fig-8 knot is chiral. Basically, you get right and left-handed versions that are mirror images but not identical.
The re-threaded version introduces an additional variable. This means that perfectly dressed re-threaded Figure-of-8s exist in FOUR distinct geometries. Therefore there are 4 non-identical ways of tying one neatly. [Add in the chirality of a stopper knot and you end up with 8 non-identical variations, so it is no surprise if you are slightly confused.]
So, in answer to your basic question:
> Surely there is a best/strongest way?
The answer is NO. Testing by Lyon Equipment for the HSE has confirmed both of the potential re-threading geometries are equally strong. See 3.3.2 of
http://www.hse.gov.uk/research/crr_pdf/2001/crr01364.pdf
However, in order to be neat and tidy, the knot needs to be both re-threaded correctly and then 'dressed' correctly. The re-threading process is crucial to whether the knot will end up 'well-formed' when it is finished. If you start re-threading on the outside of the rope you are following you need to maintain that for the 'first' turn and then re-thread on the inside for the 'second' turn (or vice-versa). You then need to tighten the knot such that the 'outer' loops end up towards the centre of the knot and the 'inner' loops are towards the ends of the knot.
So, looking at your links.
The first link is a poor demonstration and is guaranteed to lead to a badly dressed knot. The rope is re-threaded starting on the INSIDE of the existing rope, but the first turn is then made on the OUTSIDE, followed by the second turn on the inside.
The second link is initially a good demonstration of how to tie a tidy knot. They start re-threading on the outside, maintain that for the first turn and then re-thread on the inside for the the second turn. However they then fail to complete the dressing on the knots, both when using one rope and two ropes. The 'outer' loop at the top of the knot needs to be tightened so it ends up lower, towards the centre of the knot. That would then result in a well-formed neat and tidy knot.
The second video then messes things up completely when showing their so called 'finished product'. There is a massive continuity error as they show the stopper knot tied on an untidy and poorly-tied Fig-8 that is completely different in geometry from the one the demonstrated at the start of the clip.
Anyway hope all that helps.