Why does the image of cyclo[18]carbon look like a nonagon?
The first thing to say is that I'm not sure where that image is taken from; it's neither in the original article nor in the supporting information to the article. Therefore, it appears to be more of an "artist's impression" rather than an actual atomic force microscopy (AFM) image, which is what was reported in the paper.

Nevertheless, the actual AFM images of $\ce{C18}$ are in Figs. 3Q and 3R. They are referred to as "AFM far" and "AFM close" respectively because of the height of the probe ($\Delta z$):



One can indeed see that there is 9-fold symmetry (technically, $D_\mathrm{9h}$). This implies that $\ce{C18}$ has a 'polyyne' structure in which there are two different types of bonds $\ce{-C#C-C#C-\phantom{}}$, rather than a 'cumulene' structure in which every bond is equivalent $\ce{=C=C=C=C=}$ (prior to this, computational studies had been equivocal as to which form was more stable).

The bright spots within the ring do not correspond to carbon atoms, but rather to carbon–carbon triple bonds. This is consistent with the AFM images obtained for other similar intermediates in the synthesis of cyclo[18]carbon. In the authors' own words:


  Assigning the bright features in the “AFM far” images to the location of triple bonds, we observed curved polyyne segments with the expected number of triple bonds: 5 in $\ce{C22O4}$ and 8 in $\ce{C20O2}$. At small tip height, we observed sharp bond-like features with corners at the assigned positions of triple bonds and straight lines in between. This contrast was explained by CO tip relaxation, in that maxima in the potential energy landscape, from which the tip apex was repelled, were located above the triple bonds because of their high electron density. In between these maxima, ridges in the potential landscape led to straight bond-like features.


(The two bright spots outside the ring are due to individual $\ce{CO}$ molecules.)