If I have a thread of length $4$ that I want to lay on a $2\times 2$, there are $8$ ways to do it, apparently. That because from every node I can go clockwise or counter clockwise, right?
But similarly, it says that for $3\times 3$ there are $40$ ways to do it, but I could only count $20$ (like it says here). Why is that?
I assume that for $3\times 3$ you want to cover all the nodes. In this case there are $40$ ways to do it if it matters which end is the "start" and which is the "finish", but only $20$ if this doesn't matter. It's exactly the same for $2\times 2$, where you found $8$ ways because you distinguish between "clockwise from top left" and "anticlockwise from bottom left", but the OEIS says $4$ ways because it counts these as one way (which you can follow in either direction).