(written from a Production point of view)
In 2001, a new Klingon ship was called for, intended to serve as Vorok's battle cruiser in "Unexpected", season one's fifth episode of the new series, Star Trek: Enterprise. The Enterprise team, understandably exhausted after all the work they had put into the Enterprise pilot episode, were asked to do yet another design at the last minute. However, despite the work put into the design and the built of a new CGI model, it ended up not being used. The result was that a K't'inga-class CGI model was employed for Vorok's battle cruiser that was originally constructed for Star Trek: Deep Space Nine, causing a continuity error, and which had also appeared in another subsequent continuity error in the Star Trek: Voyager episode "Prophecy", as a D7-class battle cruiser.
Designer John Eaves dutifully embarked upon his task, labeling his design a "Klingon Battlecruiser "retro"", as the the D4-class designation was not yet conceived. Eaves has stated on his design process,
"I'm trying to retro the old Klingon battlecruiser. I took the old battlecruiser and took it back in time to the point where it's kind of held together by wires – kind of like the Golden Gate Bridge's technology. That long neck I thought at that time might be unstable so l got kind of a 3-piece pipe neck with heavy cables attached to the ship. Same with the engine – I didn't want framework because you see that so much so I went with cables: I thought this might be a neat thing and it has kind of a prehistory look to it, a real architecture that doesn't say framework. I just turned in the first pass on that and I'm waiting to see about it.
"In the script they kept calling for a retro warbird and we worked on that and it looked really good: it had a lot of motion on it. With the computer world now you can do all sorts of movement on a model which was discouraged before because to do motion control just required too much engineering to build into a model.
"You don't want to detail it like they did on The Motion Picture which was easy to do to bring it forward. It almost seems like that would have come before the smooth design – if you think about the whole series of the TV shows and movies you'd think that the really heavy detail look would be prior to the smooth look (of the '60s TV show ships). I tried to keep it smooth but segment the pieces so they're definite – this wing is a piece and this engine strut is a piece – as opposed to coming up with detail lines that went across the top. You see where this section is attached to that section and you see how these cables are designed to hold the piece together so it won't break off. So without going too heavy on the line work l changed the look of the engines and the way they work so it looks more industrial and there are more exposed pieces which you could easily cowl over later and you'd have the TV version." (Star Trek: Communicator issue 136)
A CGI model was constructed at Foundation Imaging, and it was during the building stage that the designation "D4-class" was coined.   Yet, the model was overruled, the producers having stated as reason for doing so that "its windows weren't prominent enough", a decision Eaves was very unhappy with.   Due to time constraints, it was decided to reuse the Deep Space Nine K't'inga-class CGI model. Foundation Imaging's Rob Bonchune, who was responsible for rendering and lighting the model, recalled the situation at the time, "It was kind of the same shape as the original Klingon battle cruiser; just a little more primitive. The way John had done it was very much like the original series one, so it had very few windows, and they were small and red. So, when I dropped the ship into the scenes that we had worked on, you couldn't see that it had windows. At the time, it never dawned on me that this would be an issue, but I understand why it is, because in STAR TREK if there are windows they are obvious." (Star Trek: The Magazine Volume 3, Issue 9, p. 46) At that time Bonchune was not quite free to deviate too much from the franchise's official position, so his remark was somewhat noncommittal. Yet, several years later, when all prime universe live-action productions were abandoned by the franchise, Bonchune, now no longer under the purview of studio policies and positions, vented his true dimmer feelings on the subject matter in much stronger wordings,
"The only other original design that was also chopped (that I remember now) was the John Eaves Klingon D-6 or D-5 [sic] that I included in my calendar image for 2006. It was originally done FOR FREE for Star Trek: Enterprise by Koji [Kuramura], who stayed up 36 hours to do it for the show. It looked great, but then the "producer(s)" said, "put more windows on it". We said "no" (you have to understand that we did so much extra, that at that point it was the straw that broke the camel's back when they were being mindlessly trivial and unappreciative). So, in their infinite wisdom, they choose to use a low-resolution K't'inga model (from a timeline over 100 years later) we had lying around. Because that was much more logical than a ship that needed 10 more windows that no one would EVER notice!!!!" Giving even further voice to his frustration over the decision, Bonchune has additionally stated on another occasion, "We all loved it over at Foundation and our friend Koji built it for free. Amazingly, even though it was a freebee for the episode, certain people in production still found a way to nit pick certain things and refused to ultimately use it until windows were added in certain places. We refused, on principle, as Koji had not slept for days building that on his own and they knew it... so instead of using it, because of, I think 5 windows that you would never see, we ended up using the K'tinga, which was UTTERLY out of place and out of continuity in the Enterprise era. Ahhh producers..."  Nevertheless, Kuramura himself expressed great pride in his build, "I had the great pleasure of building this model for Enterprise. But for reasons that were never clear, they decided to go with the older version of the Klingon ship that was based on the STMP Klingon ship. Which Jose [Perez] built. This was one of my favorite models that I got to build." (X)
As for the design itself, Bonchune has commented, "I guess my least favorite thing on it is the engines. They look like they are from Picard's time, not pre-O[riginal]S[eries]. However, the rest of it has that bulky primitive Klingony feel that I think would work. Always hard to reconcile today's FX abilities and expectations with trying to stay true to something looking more primitive than TOS."  , adding on another occasion, "Well, if we had aired the ship, I was going to refine the wings to be a little less "blunt trauma" to the aerodynamics. But when we got nixed, we moved on.....so, it stands as is."  Apart from the August spread in the 2006 calendar and the other calendar appearances, the model also found its way on the back cover of the derivative Ships of the Line book. (X)
Years later, in 2008, Eaves resubmitted his design as a possible contender for use as the Klingon Warbird in the 2009 re-imagined Star Trek, set in an alternate reality. While the design as is, was not used in the movie, its general lines were followed by Visual Effects Art Director Alex Jaeger. (Star Trek - The Art of the Film, pp. 56-57) The class designation however, was resurrected for the alternate reality D4-class, an entirely different design, in the 2013 Star Trek Into Darkness follow-up feature.
The most ironic (or cynical, depending on one's point of view) coda to the whole episode however, had already occurred previously for the second season episode "Judgment" of Enterprise; for that episode writer David A. Goodman had very much wanted to use the Klingon D7-class, but Executive Producer Rick Berman decreed an earlier model of craft was called for, as the D7 class had been depicted in the later-set Original Series. (ENT Season 2 Blu-ray-special feature, ""Judgment"-text commentary"/podcast) Berman therefore came up with the name "D5-class", and as such a new model was designed by Eaves and constructed by Pierre Drolet at Eden FX, the successor of Foundation Imaging.  It has lent credence to Bonchune's firm belief that management had originally wanted to put the creative staff into their place, or as he had put it much later, "Ahhh Producers.....you showed those fans who's boss didn't ya?"  (see also in this regard: Star Trek production staff "Trekkies")
As to underscore the fondness he had for his build, Kuramura revisited his model and had it prominently featured in his "Kobayashi Maru" October spread entry for the officially licensed 2009 Ships of the Line calendar edition. Likewise, Douglas E. Graves also built a CGI D4-class model, and had his model featured on the cover of the 2014 edition and as his April spread entry for the 2015 edition. While officially licensed, the calendar series is otherwise considered non-canon.
Cryptic Studios reused the design for their Koro't'inga-class battlecruiser in Star Trek Online. Their designation was clearly a contraction of the Koro-class, the nomenclature intended for the Klingon vessels in Star Trek: Phase II (Star Trek Phase II: The Lost Series, p. 64), and the K't'inga-class.