Morgan said he was told by the on-field umpires, Kumar Dharmasena and Tim Robinson, that they thought Stokes was not out - and gave a 'soft signal' to the third umpire Joel Wilson - but Wilson felt there was conclusive evidence to uphold the Australian appeal.
Morgan added he would have withdrawn any appeal if he had been put in such a situation, a reaction Smith called "disappointing", and there was certainly tension between the captains at the conclusion.
"It would have been a lot different if we were fielding," Morgan said. "The guy throws the ball in your direction from five yards, and all you can do is flinch. He was given out. It would have been a lot different if we were fielding - I think it was a natural reaction to protect himself as much as anything else.
"My interpretation of it was that his reaction wasn't deliberate. Kumar told me that they didn't think it was out ... and the third umpire has disagreed. I feel the ball was thrown so fast you can only react in a way that defends yourself. I think he put his hand up to protect himself, and followed the ball a little bit. How you can interpret that is open. But certainly, I didn't think it was deliberate."
Morgan, however, said he did not suggest to Smith that the appeal should be withdrawn. "He's entitled to appeal. If he thinks it's out, he's going to appeal."
And Smith was in no doubts over his actions, an early tester in his young captaincy career, although the appeal was instigated by Matthew Wade behind the stumps.
"Wadey had a good view and said straightaway that he thought the ball was missing Stokes and hitting the stumps, so we appealed and it went upstairs and the umpire gave it out. The way I saw it was that he was out of his ground and he willfully put the hand out - which is rule I've been told - and he got given out by the umpire."
"The umpires are there to do a job, to make a decision. It went upstairs to the third umpire, and he saw it the same way we saw it - and it was given out. If you willfully put your hand out in front of the ball then you're given out, and that's the way it went."
With a short turnaround between matches in this series - the sides meet again in two days at Old Trafford - there could be some residual animosity from England, although Morgan suggested the end-of-match exchange had laid all the views out there.
"I gave my thoughts, and he gave me his. It's nothing big. I don't think it was the winning and losing of the game. So it's not a big deal, just his view against mine."
Smith said: "I was just saying he was out of his ground and he put his hand in the way of the stumps. I put it into perspective and said it was the same as me coming back for a two and turning around and putting my hand out. It just looked worse because it went back to the bowler and it all happened so quickly.
"If you look at it, from what I saw, the ball was going towards the stumps and wasn't going to hit him. He's put his hand out to stop the ball."
For Stokes, it was the second time this summer that he has walked off Lord's after a curious dismissal against Australia. In the Test match he was run out in the second innings when he jumped to avoid a throw from Mitchell Johnson instead of grounding himself over the crease.
This incident made him just the sixth batsman to be out obstructing the field in a one-day international and only the second England player in any format after Len Hutton in a Test against South Africa, at The Oval, in 1951. That remains the only such occurrence in a Test.
Andrew McGlashan is a deputy editor at ESPNcricinfo