Entering Japan Documents

I am triple vaccinated and can prove so with the NHS application. However, it seems to me that this is not sufficient enough (or even accepted for that matter) for entering Japan. Upon reading the the entry requirements for Japan on the UK's government website, I read this: "As part of travelling to Japan, you must complete information on the MySoS app or website, including signing a written pledge to abide by the Japanese rules on self-isolation and other COVID measures."

Straight away, this confused me as I do not understand the diference between the MySoS app and the "fast track" website that Japan seems to be using. The fast strack website link is below.

https://www.hco.mhlw.go.jp/en/
https://vjw-lp.digital.go.jp/en/

I do not quite understand what I need to fill in in order to come to Japan. Is the MySoS app still being used? I have filled out the 1st step shown on the second link, however as I have not yet booked a flight to Japan, I cannot continue with the application until I have done so.
Where does the "fast track" website show my vaccination status?
I am very confused by all of this, and I've read it through several times. I would much prefer to just show my NHS QR codes for vaccination status, but I don't think it is accepted.

That UK website is out-of-date and was inaccurate even in the past. MySOS is not accepted anymore and was never mandatory. For up-to-date information, check this video:
https://www.youtube.com/watch?v=P0L7x1XMg_A