The Right Way to DM

I had a comment on Communicating with the Players where the reader thought I was prescribing a 'right way to DM' that was at variance with the way everyone has done it for 35 years.  Well, I first DMed in '77 and have done it many different ways.  I never have said which is the right way. But since you ask....

Let me start by clarifying what I meant by skill rolls overtaking role playing to the detriment of the game.  Here's two variations on a scene.
While the rest of the party fishes the fighter out of the pit, the elven M-U/Thief announces he's going to search the rest of the room for traps. A clear statement of action.

Example 1
Player: I roll a 18.
DM : You found one.
Player: 20 on Disable Device
DM : OK, you disabled it.

Example 2
Player: I roll a 18
DM : You find a suspiciously square area on the floor.
Player: I prod it with my staff.
DM: It gives a little and sounds hollow.
Player: 20 on Disable Device
DM: You noticed that the arm of the adjacent statute started to rise as you prodded the floor. Unlike the rest of the statutes in the room, this one's shield is on the floor.
Player: I'll replace the shield
DM: It's made of stone, you'll need a strength check to lift it by yourself.
Player: I rolled a 7, maybe I'll just mark the floor so no one steps on it.

  The first example is in accordance with (Pathfinder) rules; "If the check succeeds, you disable the device." The character's actions are completely abstracted into a die roll.  The player has no agency.  The second is the way I run the skill, success tells you how it can be disabled (immobilize the arm), the player has the ability to decide how the character accomplishes the action. Or to think out of the box for another way to remove the threat.  I know which I prefer, but your mileage may vary.

Rod's Rules for DMing

Actually you just need to answer two questions to determine if you are DMing the right way.
Are you having fun?
Are your players having fun?

If everyone is enjoying themselves you're doing fine. Save agonizing over doing it 'the right way'  for when you're involved in a land war in Asia or engaged in a battle of wits with a Sicilian with death on the line.