Although the websites that you provided links for have good information about applying for ETA for travel to the United Kingdom (UK) I don't think that they are the official government website.
I believe that this is the official UK government website:
https://www.gov.uk/guidance/apply-for-an-electronic-travel-authorisation-eta
Currently, US citizens don't have to apply yet, but that could be changing sometime before the end of 2024. You can sign up for email notifications when they update the list of countries that need to apply for ETA.
I am cruising out of Southampton in June, 2024, too and just keep checking the website occasionally.
Hopefully someone from the UK will comment on this thread.