So, there are you are, at the retaining wall for the Temple courtyard, about as close to the Holy of Holies as a Jew can get these days, davening Shacharis (morning prayers). You hear something from the other end of the mehitzah (separation between men and women) - possibly women reading a Torah scroll (it is a custom that women not do this when men are present).
Unwittingly, perhaps out of curiosity at first, you focus on that sound, that irritating kernel of imperfection in your environment, how it's forcing a loss of your concentration in tefilah (prayer), draining the holiness from your avodah (service). You would protest verbally, but you're already into Psukei Dezimra (after which no other speech other than prayer is permitted), and anyway the women never listened to you when you asked and demanded this in the past. It gnaws at your mind as you begin the blessings preceding Shema (a very significant prayer for which absolute concentration is needed). How could they allow such a thing in this place, you think to yourself. I'm almost at the Shema, and I can't even focus. This is against Halacha (Jewish law). It is indecent. Someone has to do something! So you ask yourself, tzadik (righteous person) that you are, what would Pinchas do?
When a sentence is carried out after the due process of a trial and conviction, there is less of a need to dwell on the motives of the judges and executioner: they're going by the book, and we can check their behavior against the book. But the motives of the zealot who takes unilateral action are extremely important, for his very qualifications as a zealot hinge upon the question of what, exactly, prompted him to do what he did. Is he truly motivated to "still G-d's wrath", or has he found a holy outlet for his individual aggression? Is his act truly an act of peace, driven by the desire to reconcile an errant people with their G-d, or is it an act of violence, made kosher by the assumption of the label "zealot"?Whoever it was that thought themselves a zealot by throwing chairs at people, they were arrested a short time later.
The true zealot is an utterly selfless individual -- one who is concerned only about the relationship between G-d and His people, with no thought for his own feelings on the matter. The moment his personal prejudices and inclinations are involved, he ceases to be a zealot.