I think that this is basically an accounting problem. The full electron configuration of $\ce{Ne}$ is: $1s^22s^22p_x^22p_y^22p_z^2$. So, what does one now define as core electrons? It could be: "all electrons not in the outermost shell", yielding the $1s$ electrons. An alternative is "all electrons in completely filled shells", yielding all 10 electrons as core electrons. (Note that these two definitions yield identical results for almost all non-noble gas elements, with some difficulties surrounding the $f$-block.) Both definitions have some merit, I would prefer the first one, giving an effective charge of $+8$, although I do not know what one does with that number next.
A word on effective nuclear charge
As noted on the wikipedia page linked in the question, to arrive at more meaningful effective nuclear charges (as might be helpful for the systematic creation of basis sets for computational/quantum chemistry), one has to take the effective shielding strength of the different orbital shapes into account. The $d$-block-contraction, as can be seen in the somewhat different chemistry of $\ce{As}$ compared to $\ce{P}$ and $\ce{Sb}$, is due to this. Thus, in the end, a pen-and-paper calculation as alluded to above can be useful to think about some concepts in a qualitative fashion, but more extensive calculation is needed for quantitative reasoning.
