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From the SRD:
Lawful characters tell the truth, keep their word, respect authority, honor tradition, and judge those who fall short of their duties. Chaotic characters follow their consciences, resent being told what to do, favor new ideas over tradition, and do what they promise if they feel like it.
"Law" implies honor, trustworthiness, obedience to authority, and reliability. On the downside, lawfulness can include close-mindedness, reactionary adherence to tradition, and a lack of adaptability.
"Chaos" implies freedom, adaptability, and flexibility. On the downside, chaos can include recklessness, resentment toward legitimate authority, arbitrary actions, and irresponsibility.
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Lawfulness does include respect for legitimate authority. One who violates the legitimate law of the land in pursuit of good, when other options for pursuing good are available, is IMHO, NG. One who does so consistently, and withou regard for legitimate authority would be CG.
The SRD:
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Good characters and creatures protect innocent life. Evil characters and creatures debase or destroy innocent life, whether for fun or profit.
"Good" implies altruism, respect for life, and a concern for the dignity of sentient beings. Good characters make personal sacrifices to help others.
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Goodness includes a respect for life, and a respect for sentients that is, IMHO, incompatible with slaying such a sentient merely on the sayso of a first level spell that gives no specifics of a beings transgressions.
One who consistently kills without regard for sentient life - especially if the motivation is not altruisitic, but, rather, adherence to orders or a church's code, would IMHO, be evil.
There are degrees of evil, and not all of them merit a death sentence.
If the cleric in question would strike down the child who detects as evil because he consistently and maliciously steals items of low value from a local merchant, I would not call him lawful good.