So if I read that right, the SWR is not necessarily affected by which side the coax center is connected too, but the radiation pattern is, and is better if the center is attached to the long side?
Correct! Part of the radiation pattern problem with a J-pole is that there are multiple balance problems:
1) Unbalanced coax attached to a balanced quarter-wave transformer.
2) Balanced quarter-wave transformer attached to the very end of a half-wave radiator which at that point is an unbalanced connection.
Common-mode issues (the outside of the coax braid acting as part of the antenna) are what distorts this pattern, and is one reason you see most of them choked near the feedpoint. While not perfect, the center conductor of the coax should go to the wire of the balanced transformer that ends up connecting to the half-wave radiator (basically the long side).
The radio won't care which way you connect the coax, but if the coax center conductor is connected to the short side, the common-mode current in that configuration tends to make the radiation pattern point downwards.
For those with repeaters close by, or casual simplex operations this may not matter much, but most want to do it right. Still, the common-mode braid radiation issue is one reason you never see J-poles in the commercial world, unless perhaps the guy running the place is a ham and decides to put one into service anyway.