Eric, I think you analyzed it perfectly, I agree with every word. (I also agree with many of the things others have written in this thread.)

When you speak of the "Real Columbo underneath" you hit on one of my favorite things about the early shows. It was very clear from the first episodes (and the two pilots) that Columbo was two different people: one to the murderers and one to everyone else. When he knew that someone wasn't involved in a crime, he would play things fairly straightforward. He didn't act like a forgetful bumbler. He was tough and matter of fact. He would also drop the bumbler act when the murderer knew that he knew they were the guilty party and saw through his act. Then he would just be himself. I liked it when Robert Conrad's character,Milo, confronted him in the hospital waiting room:

Milo: "Your health program didn't last very long."

Columbo, "Long enough."

Milo: "You know something, Columbo: You're a devious man."

Columbo: "That's what they tell me."

I loved this aspect of his character!