O'Brien is also wearing (two) lieutenant pips on his dress uniform during his wedding in TNG's "Data's Day."
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Every picture tells a story!

And yet, despite wearing Lieutenant's pips in Family, Sergei Rozhenko instantly recognizes O'Brien as a Chief Petty Officer.
Either Sergei Rozhenko had a bad knowledge of Starfleet ranks or he was deliberately insulting Miles.

I did all sorts of back flips and contortions many many years ago for a website I created trying to make sense of O'Brien's pips in TNG. In the end I ignored the pips and stuck with the DS9 chevrons.
I can see your point. But I'm a nitpicker who also loooove to come up with somewhat complicated explanations for certain errors and mysteries. On the Kes Website, I have made a page about all those errors and contradictions in Voyager. I may have to expand it to DS9 and TNG as well.

Yes, the photo of Miles on the Memory Alpha article if you blow it up clearly has two additional dots to the right of the chevrons. They're smaller than the officers' pips, and not really a good design of insignia - you shouldn't have to get up to a foot from the man's collar in order to see what rank he has.
Clearly just inconsistent wardrobe choices. Though if you feel like you must give it an in-universe explanation you could say he was given an acting commission as lieutenant for a short time in order to be an acting section head or something and then decided he'd rather let it expire and revert to his permanent rank as an NCO.
Sisko was less of a by-the-book CO than Picard and was fine with O'Brien having just operational authority over the few starfleet engineers reporting to him.
Yes, it is two small dots. I wasn't really sure what they meant but I found a picture with Starfleet ranks on Pinterest (see below) where everything is explained.
Obviously those Petty Officer ranks with chevrons and dots have replaced the Warrant Officer rank in Starfleet.
However, it doesn't explain O'Brien's Lieutenant pips in TNG and why he lost them.
"Wardrobe error" might be the Gray Universe solution.Not according to dialogue in several dozen episodes, which refer to him as a "Chief" or "Chief Petty Officer" not a Lieutenant (which was a single unscripted occasion) and a likely wardrobe error.
But I would like to have a Star Trek Universe Solution as well.
Oh, yes, he was! (Oh yes he was!), Oh, yes, he was! (Oh yes he was!)Was he?
He was a Lieutenat on the Enterprise for four years
Then he was demoted and no one knows why
Such a dedicated Officer in Starfleet!
Sorry, I just had to do that! Please listen to "Deicated Follower Of Fashion" by the Kinks which gave me the inspiration for that little tune.

Could be, or it could just be a placeholder for a "subordinate officer" ranks, given that he continued to wear it on his dress uniform as a Senior Chief Specialist (which would be the equivalent of a warrant officer in the British sense, but not a US W-1)
Note my previous comment about the Warrant Office rank and that may have become obsolete in later years in Starfleet.
No, because he's going from at most a section leader to a Department Head, therefore if anything even if he was a Chief Petty Officer on the Enterprise, he should have been a Lieutenant by the time he was prompted to Senior Chief Specialist.
Then he was promoted in role but obviously demoted in rank when he arrived at the station Deep Space Nine.
Nothing, because there was no demotion after or during his transfer as he had the same rank and a promotion in role on the station compared to the Enterprise.
He was wearing a Lieutenat's uniform for four years on the Enterprise and then he becomes a Warrant Officer (and later on a Senior Chief Petty Officer ) on Deep Space Nine. I just get the impression that he was demoted between the events in Realm Of Fear (TNG) and Emissary (DS9).
Something must have happened and I have a feeling that Riker was involved.

Really?
Last time I checked a Sergeant's insignia has three chevrons and zero additional elements (rockers, stars, pips)...
Whereas O'Brien's insignia has three chevrons and two additional elements, which would put it as a Senior Chief Petty Officer (which closely matches his stated rank of Senior Chief Specialist).
I stand corrected here.
I found this on Pinterest which explains everything about this insignia.

For everyone who reads this, especially fans of Miles O'Brien:
I'm not out to mess with and scandalize O'Brien in any way. He's a great character and definitely one of my favorites.
The same for Riker. A character I also like.
I just like to come up with explanations and solutions for odd things and contradictions in Star Trek.
When it comes to O'Briens ranks and the contradictions we see on screen, it was just too good to miss. It could as well have been about Bashir, Jadzia or Ezri.