Modify spark igniter circuit for bug zapper functionality?
This question is ambiguous, which is perhaps why it has not been answered.
A spark ignitor is designed to produce spark when the power is applied, which is accomplished by generating sufficiently high voltage to break down the dialectric between a spark gap - every time.  A spark igniter that did not spontaneously spark would be a failure for the purpose of causing ignition.
A bug zapper is designed to apply high voltage to a grid that is designed to be less than the voltage needed to break down the dialectric between the gap, so that it is not just continually sparking, which would be an undesirable nuisance.  But when a bug flies or is scooped to be between the gap of the grid, its body artificially bridges the gap to be less than what is needed for a dialectric breakdown, which allows the spark to take place and zap the bug.
Your question about adding the output changes to convert the ignitor to a zapper, goes against your stated use-case of creating spontaneous arc (discharge) for art.  But, in an attempt to answer your question, the design difference that the output diodes in the zapper bring to bear, is to rectify the output to be pulsed DC which then can charge the additional HV capacitor to allow for a higher current capacitive discharge into a bug when they bridge the gap. The resulting spark from a capacitive discharge would be with greater current, resulting in more effective kill strength, I suspect.
But to the question of polarity of the output, it is also kind of a misnomer because the oscillator of both circuits produces a pulsed DC rather than a true AC input signal which is essentially a repeated half-wave. This creates only single magnetic polarity on each cycle which will also only create single polarity flyback voltage in both circuits.
The existence of a spark or discharge is solely based on breakdown voltage for a given spark gap and whether the circuit meets or exceeds this value.  Polarity and current are additional characteristics that may be concerns or affects that also could be maximized or minimized depending on desired purpose. FWIW