ColinWright's point is that drawing such an arrow on a tiny circle is equivalent to drawing the letter R on the 2D manifold. It gives a local orientation to the 2D surface. (If you wish you may think of this as an arrow into some 3D embedding space, but you don't have to.)
If the 2D space is orientable, then when you take a copy of this little circle (or letter R) and go for a long walk, when you get home your copy will always match the original. That's all that orientable means. In the standard usage, it's a property of the 2D manifold, not of the particular walks you take. I think this is the point of confusion here.