Ok here's another summary from my part:
First of all, I think your wording about what makes a cut significant could be clarified in a better way. You mention "...which omits at least a single verse, chorus, solo, or movement in the song". I think all these can be summarized with the term "section" which is similar to "movement" in music theory anyways. A verse, chorus etc. is a section so it would fit for all of those. Also this should probably go into the glossary for "cut", not the rule itself I think.
In the OP you also say "The problem here does not arise for official cuts, as in the ranking criteria it is stated that, for example, a TV-size beatmap should be specified as such in the title, however fan-made works are not included." but this is not quite true as any other official cut that is not TV Size will not necessarily have a marker which indicates it.
You could also simply remove that thing about Harumachi Clover since it was just a misunderstanding and has already been explained.
I'll write my main arguments down again for better visibility since the thread got cluttered:
1) Unlike (Short Ver.) and (TV Size), the (Cut Ver.) marker doesn't really exist in official sources so enforcing it onto many maps would be a significant alteration of the original metadata and that goes against the idea of the following RC rule: "Do not modify the metadata an artist provides on official sources unless said modification is done in order to comply with formatting and standardisation rules on this Ranking Criteria."
Of course this would fall under the exception mentioned in the rule, but keep in mind that a cut doesn't change the way the song sounds at all, therefore it is not an actual edit/remix, just a shorter version. That should not justify the introduction of an artificial marker.
2) Since all official cuts except TV Size songs don't require a marker unless it's already in the metadata source, enforcing (Cut Ver.) would result in official cuts to have the same exact title as the full version and thus still be misleading. If (Short Ver.) etc. isn't being enforced, neither should (Cut Ver.) be.
3) A song can be cut in many different ways and different lengths so if for example someone makes a 1 minute cut of a song, it would still have the same title as a 3 minute cut. This can be an even bigger problem if a song consists of parts which sound very different from each other. Some known examples for this are
https://osu.ppy.sh/s/823960 and
https://osu.ppy.sh/s/456366 but there are actually many songs with such a contrast (mostly electronic songs with a calm intro/outro and very heavy drops). Cutting the song in a way that makes only one part of the song remain would make the Cut Ver. marker apply, but if someone else cut the other part, it would get the marker as well despite being completely different, so the marker wouldn't make it any less misleading.
4) What if someone cuts a song but extends it at the same time? Like, if one part of a song is removed but a different part that happens to be longer is being looped, it would be both a cut and an extension at the same time. I'm not sure if this has been done before but considering all the audio edits people have done in order to go around song length limits (adding R3 Music Box, looping parts of the song several times, putting 2 songs together etc.), this would be a viable possibility.
The RC rule "The audio file of a song should not be artificially extended in order to meet a time limitation in the beatmapset section of this criteria. This can include (but is not limited to) looping sections of the audio file, lowering the BPM of the song or section of the song, or adding small amounts of music to the song without incorporating it throughout the entire song. This does not apply to song compilations or audio files less than the minimum rankable beatmapset length." only discourages it if it's done for the sake of reaching a certain drain time threshold, but if that is not the case, it's fine to do so.
5) If a song has a long outro (more than 20%) which the mapper doesn't want to map, they would have to cut it for the sake of rankability, but then they would have to add (Cut Ver.) to the title despite basically mapping the full song. And also if it's less than 20%, it is not rare that mappers decide to not map the outro, which is essentially the same as cutting it. This would cause another contradiction where two maps that have the exact same parts mapped have different metadata.
6) There would be more metadata related things to check for BNs and QAH (a lot of them don't check the metadata in the first place) and it could be difficult/confusing if there are different versions of the same song or if there are no official uploads/sources where the song length can be taken from.
7) Having this marker on a big portion of ranked maps is simply not very pleasant and makes the song title unnecessarily longer. It would be even worse for songs that have (Sped Up Ver.), (Nightcore Mix) or (XY Remix) etc. in the title because many of them are cuts so those would have 2 markers next to each other and that looks horrible and takes up many characters (there is a character limit).
Edit: One thing that also came to my mind is that it would also be a contradiction to have stuff like (Long Ver.) (Cut Ver.) in the title for songs such as
https://osu.ppy.sh/s/705224,
https://osu.ppy.sh/s/522725 or
https://osu.ppy.sh/s/557145 and also for (Full Ver.) (Cut Ver.), for example
https://osu.ppy.sh/s/473048,
https://osu.ppy.sh/s/801506 or
https://osu.ppy.sh/s/6798768) The main issue seems to be that some players are disappointed when they realize that the map they downloaded is shorter than expected, which implies that the length is the deciding factor (as confirmed by someone in the thread). To fix that problem, changes to the beatmap listing and osu!direct need to be made. The information about whether it's the full version or not doesn't seem very important to me when downloading the map/playing it. If someone is specifically interested in a certain song, they will most likely try to find it on the internet and probably discover its full length there. If anything, it could be included in the tags/beatmap description which would still be visible for people who download the map without modifying the metadata. And no, the beatmap listing isn't supposed to include every single information about every map, that's what beatmap pages exist for, if you want to know more about a map you should simply click on it.