a crazy psycho killer is chaotic evil even when hes nice to his family.
a paladin who kills criminals still s lawfull good.
Not crossing the street at a designated crosswalk is a crime. Parking your cart in a way which blocks the street is a crime. A child shoplifting a tiny wooden toy horse is committing a crime.
In the world you and I live in, not all "crimes" are grounds for summary execution on the spot. The same goes for the world of D&D. You need to be very, very careful about what your paladin defines as a crime worthy of death. Without concern for justice, without concern for GOOD, your "paladin" is not lawful good in any way at all, but neutral (or evil). Under your definition, the "paladin" is the same as the crazy psycho killer.
No good DM would let a player at a tabletop game get away with a paladin summarily executing someone for stealing an apple and still claim to be lawful good.