I would try to aim for the following intuition: Your conic is either non-degenerate or degenerate, which is what your discriminant decides. If it is non-degenerate, then over the reals you'd get something that is topologically equivalent to a projective line. You can define a projective basis on it and describe every point in a unique fashion with respect to that basis. All of this should carry over to finite fields just fine.
General idea is that if you take a point on your conic then there are $p+1$ lines through that point. Each of them will intersect the conic in one well-defined other point, except for one line, the tangent, where that second point is in fact the same one you started from. So the $p+1$ lines through the point correspond $1:1$ to the $p+1$ points on the conic.
For the degenerate case over the reals, you might want to distinguish three cases. Either your conic is a pair of real projective lines, or a single line with algebraic multiplicity $2$, or a pair of complex conjugate lines which intersect in a single real point. Transported to finite fields, the first of these cases would yield $2(p+1)-1=2p+1$ points since the point of intersection is only counted once. The third case yields $1$ point. The second case again yields $p+1$ points, so at first glance I would expect that to be an option as well. But since your formula is not fully general due to the absence of mixed terms for $xz$ and $yz$, it can't express duplicate lines while maintaining $d\neq 0$.