Personally, I think there's a subtle yet important distinction here. "You examine the chest and find no traps" and "You think it's not trapped" are different. In the former, the DM is narrating the result of the adventurer's actions. In the latter, the DM is telling the player what conclusions his character draws from having found no traps. How the DM communicate this matters, especially as it relates to what the player chooses to do next. This is very similar to how I see some DMs narrating the result of an adventurer's action when the player fails a Wisdom (Insight) check. Commonly, DMs say "You believe him." I think this is misleading and oversteps the role of DM. "You're unable to discern if he's being truthful" is better.
Having said that, I don't typically narrate the adventurer's action in the wake of the player failing a check in this way. I much prefer to reveal the trap but escalate the tension by having their search put them in a spot or cost them a resource like time. Doing it this way also sidesteps a problem commonly reported by DMs: That players "metagame" by hearing "You examine the chest and find no traps" while a looking at a low roll of the die. This can lead to players continuing to search repeatedly which often results in DMs making these checks in secret for the players so they don't respond in that way. (Which of course leads to another problem of players basically playing a guessing game now.) This can all be avoided by simply revealing the trap after a failed check and then complicating the situation or making it cost.
At least, that's how I see it. It should go without saying that anyone else should do what they want to achieve the goals of play.