Reasoning: Change in common name used by independent, reliable, English-language secondary sources, per WP:COMMONNAME – Wikipedia does not necessarily use the subject's "official" name as an article title; it generally prefers the name that is most commonly used (as determined by its prevalence in a significant majority of independent, reliable English-language sources) over the last 12 months. Note, COMMONNAME does not mean most popular name per Google Trends, as stated in WP:WIAN, but the COMMONNAME in sources. As stated in the supporting evidence below, the lead of WLS in Ngrams has drastically reduced, with Welsh forms sometimes taking the lead, and Google News Articles from the last 12 months show more use of the Welsh name. Manually searching shows many publications in the UK and Wales use the Welsh name first or not in parenthesis, while possibly using WLS secondly, in parenthesis or as a potential descriptor. International recent sources are scarce, while many use WLS, they discuss past events which per WP:NAMECHANGES and WP:WIAN shouldn't be given too much weight in terms of context.
Sources which use one name first or uses one in parenthesis are counted as preferring one name over the other. If WLS is used after CyI, it is classed as a potential descriptor, even if capitalised.
Italics: use different names in different articles; excluding publications using either Welsh name. Strikethrough: does not discuss strictly current (~10 yrs) events per WP:NAMECHANGES, such as using Constantinople when describing the medieval period. Underlined: Uses one name solely, no alternative present.
Google Ngrams – WLS was preferred primarily in the past, but some recent years have CyI ahead. For 2019, combining the two forms of the Welsh Name, over takes the small 2019 lead of WLS.
Google News results – Sorting per the last year, "Cymdeithas yr Iaith" produced 10 pages of news results, while "Welsh Language Society" produces 6 pages. Note, including articles more than a year old does lean more towards WLS, but limiting to recent results (~12 months), there is a majority for the Welsh name.
✗ Fail – Proposal failed, avoid the argument of primacy, if the existing name is used in anyway, the argument is invalid. Although later opposition comments were borderline nonsense in regarding policy.
The University formally announced in April, it would be dropping Glyndwr and undergoing a rebrand. It's website, Wrexham.ac.uk is now for the new name, previously being glyndwr.ac.uk. News sources, such as the recent controversy on a visiting professor, fully independent on the topic of the name change, are using the new name. These include, BBC News, Nation.Cymru[31], Daily Post, The Leader, WalesOnline. Other sources on other topics also use the new name, such as Business News Wales, and the local council. Therefore this should statisfy, WP:NAMECHANGES, Sometimes the subject of an article will undergo a change of name. When this occurs, we give extra weight to independent, reliable English-language sources ("reliable sources") written after the name change. If the reliable sources written after the change is announced routinely use the new name, Wikipedia should follow suit and change relevant titles to match. as multiple news sources after the name change routinely use the new name over the old. With the old name being absent in these sources.
Reasoning: Change in name used in multiple sources, per WP:COMMONNAME which states prefers the name that is most commonly used (as determined by its prevalence in a significant majority of independent, reliable English-language sources). WP:NAMECHANGES can somehow also apply per the 2012 discussion above stating the infrastructure operator Network Railuses the term[32] if determined as the time of an (official) name change, and now many reliable secondary sources have since followed suit. Such use of the term "Main" over "Coast" was greatly apparent and increased following recent announcements, which overwhemingly used "Main" over the last 12 months. Google Trends also backs the change, although Google has issues per WP:WIAN, while there is not enough data for Ngrams and Google Search can be skewed by feeds and the game dlc.
Prefer "Main Line" as separate words for consistency with West Coast Main Line and South Wales Main Line. Because of the sheer use of "Main" in the recent announcements, I raise this RM to at least test the waters, but some still use "Coast" and this can be skewed on recent events, but it was overwhemingly for "Main". Plus "Coast Line" can be confused for "coastline", but that is minor.
While Transport for Wales Rail uses "Coast" on their Network Map it expands to Birmingham via Shrewsbury, so more descriptive of a service rather than the subject of this article. Nonetheless is only one source and a mere passenger operator rather than owner of the line.
Other stats:
Google Trends (do not rely on alone per WP:WIAN) There are searches for Main Line and almost none searched for Coast Line over the last 12 months.
Google Ngrams no data is found except for Coast Line, so no comparison can be made, plus only up to 2019.
Google Search not performed, skewed by unrelated news feeds (for now) and the dlc for Train Simulator Classic, and general use possibly based on Wikipedia's usage, while being discouraged for other names under WP:WIAN.
Google Trends as an indicator, there was no clear commonname search over the last 90 days on Google Trends. Snowdonia Marathon had the most, but only a few searches more than Marathon Eryri, considering the very low search volumes.
Google News (July to October 2023) - Marathon Eryri 2-4 news articles, Snowdonia Marathon full of index pages so hard to decipher, but only saw 2 or 3 actual news articles (but they referred to the past or a quote), Snowdonia Marathon Eryri none.
Not approved proposal; see Talk:Welsh Language Society#Requested move 29 September 2023
Accepted proposal; see Talk:Celtic Dragons#Requested move 20 November 2023