Tierlist

The tier list itself reflects the author's opinion about how servants work and you know different people have different ideas.. However, because JP is 2 years ahead of NA so they already have Scathath Skadi, which is pretty popular at the moment (or so called meta itself), in addition to that, break bar also limits oneshot buster meta and as a result, buster-oriented servants got de-throned from their reign... But regardless, Musashi is still a very powerful Saber with very high stats and unique gimmick until this day in JP, so I don't think you need to worry much about the Tier List here, just play it the way you want. Game is even cheesed with 1-3 star servants anyway :)

No tier list is perfectly objective and bias free, and different communities/individual authors will rarely agree on everything.
It's also possible that Musashi is causing some "new toy syndrome" due to her unique dual wield, and recent release.

You can ignore appmedia tierlist for the most part, it's absolute rubbish.

Because Cirnopedia tier list is from JP server, which the meta is quick thanks to Skadi. Gamepress tier list is for NA server, we are at buster era right now, so most buster servants are high tier.

Edit: don't worry about tier lists in this game, most servants are good and useful if you know what you're doing.

All tier lists are just a reference (include Rabbit's and gamewith.ect, etc). It is good for newcomer and FTP players with very strict plan for rolling, but none of the tier lists will affect your servants' performance in your own team. The most important thing is that [There is no standard criterion]. therefore you dont really need to care too much about tier list.
Honesly I do even prefer Gamepress's tier list, it is not because I mainly play NA, but in fact I followed Appemedia for over 1 years. They are emphasis current performance on JP server, and WAY TOO MUCH. Like others mentioned above, the coming of Skadi changed almost every Quick oriented servants' performance, but if people dont have Skadi, they wont benefit from that.
ONE MOST REPRESENTATIVE EXAMPLE is Okita vs Musashi, although I dislike the nonsense Waifu war, but the fact is, before the come of Skadi, Musashi's ranking was in the middle of tier list (but generally best sabers) but Okita was bottom. Then right now they are reversed. Are there any Nerfed or Hyped of their own performance? None. Multi-bar hp already existed before Skadi's came. However, some idiots used Appemedia's tier list attack other servants and became a serious cause of Waifu war.
Same thing. Edmond Dantes, if you dont pair him of Skadi, he is still wont work good even after his improvement from his NP strengthening. Rank S in the same as Gil, best DPS? I wont deny the powerful Dantes+dual Sakdi could be even better than Gil+dual Merlin or any Art team, but does it matter to his own performance?

many thing.
different Servant make different meta.
Different enemy make different meta.
New Servant make different meta.
To be fair, i think Merlin is still better then skadi, but skadi is better for farming in is own way.

To be fair despite i absolutly live for tier list (watching and commenting tier list is my hobby) all of them fall in the same error.
they rank servant based on multiple principle and rule, instead of creating a rule set for all.
For exemple, the "rule" for T2 of 5* is
"servant who are the best in their role and can ecc ecc"
the key word is "best in the role" (there is more in reality, but for this exemple i will make it simple)
so, who is the best is role (in musashi case, saber , st buster) get the T2.
fine, but then you have the rider case.
Ozy and Kuku (quez).
they are both ST Buster servant. Only 1 for role can be there if we follow the rule. I belive Ozy is a tad better then Kuku. So kuku should be T3. But you cant have him in the same tier of Karna, who is a weaker servant.
Or for exemple Medb (who does not deserve the T6 at all.
She is so much down in the tier list despite being better then most of the T4 (and when she get the 2 buff, she can easly contend with ozy and kuku, even win if ageinst male), only becouse she have lot of competition in her role. she is the weakest of 5* rider (well, meaby iskandar is weaker then she is), but she does deserve t6?
yes, actually. becouse the rule is "servant outclassed in their own specialization" and she is outclassed by ozy (for now).

Another (more recent ) exemple is king hassan. He is the best ST assassin in the game. this alone is enought to give him a place in T2, but, becouse he is nor really usefull he got placed in T4.

Becouse of the nature of the game, i personally think the actual system of ranking is kind of missleading. We have many servant who are specialized in different area of game play, and in their niche can easly outperform the strongest servants in the game.
So, i suggest to try a different approch.
Instead of a tier list, we can use a score list, and then we can use this score to make a tier list more accurate.
For exemple merlin is complitly broken, and we give him a 100/100 score (random number) and then we can have weaver and give him a 95/100 score.
And so on and on. then made the review more focused on the ideal team play of the servant.
Then, and only then we can star to consider meta game for team comp.
for exemple, we can simply say medb is a 75/100, but with merlin, and ageins a male she is a 85/100.
Or Dantes is a 60/100, but with skadi he is a 90/100 ecc ecc.
We can even simply say, becouse this game is runned by support, thet " if X servant is with his kind of support, he get a +Y "
ah, yeah this would be nice, right?

Just to hop in here for a bit of FYI:
Cirnopedia's tier list is just a simplified version of the one from AppMedia, and it looks like Cirno hasn't updated in a while.

As others have said, it's mostly down to individual preference and the fact that cirnopedia and appmedia both heavily favour Skadi. Another tierlist made by a gameFAQs user 22whiterabbit22 favours Skadi less, so Okita ends up on a lower tier than Musashi. It's due to people favouring different play styles.

First off, do you know that appmedia have 2 different tierlists? The one that you're looking at is based on recent CQ on JP environment, and the other one is more favored on general content.