Airport Utility shows that the TC is connected (internet & TC both green).
First base.. check.
2/ In Finder try and mount the TC.. you simply click on the icon and it should mount.. make sure you set preferences in Finder to show network shares.
Did you try this before attempting the connect to server.. sorry I was unclear.. we need to know if you can see TC in Finder without needing any further work on making it appear.
Here is screenshot from my Finder. Note it shows under locations, external drive partitions, hgst1 & 2, my NAS, J1900NAS, and the Time Capsule tc5e which is not mounted.
You may need to change Finder Preferences to show all items under locations in sidebar. Apple do not turn them on by default.


I click it to mount.

I clicked "Connect to Server" & typed in my IP address & clicked "connect", but the next box you showed me (for registered user) never came up.
Second base.. fail.
Your computer is not even attempting to connect to the TC hence the fail. Hence you cannot backup.
Since you didn't post screenshots.. I cannot help beyond this point.
What I would do at this point is press the reset button on the TC with a suitable pin.. for 10sec so it resets. Then do the setup of the TC again in Airport Utility from the Mac. Use all short names, no spaces and pure alphanumerics. NOT APPLE RECOMMENDED NAMES. This is most important. Passwords same rules but longer.
No backups will be lost. This will simply fix network issues. However you will need to add disk in Time Machine to enable Time Machine to continue backing up.
Should I say "Add backup disk & try creating another one?" (I'm assuming if it doesn't work it will have written over all my current backups??)
Just to alleviate any fears.. no.. Time Machine will never write over its own backups.. it will delete them due to corruption on a regular basis.. or totally destroy them with upgraded OS.. but never just delete them.. and usually with some warning and permission from you.
As the simplest solution to your problem.. let me strongly recommend you simply buy a USB drive (with the USB type that matches your Mac) and start doing direct backups.. forget using Time Capsule. This is what I suggested at the end of the last post if the fix didn't work. It hasn't.. so make the change.
Use Cloud for backing up your user files.. iCloud being the Apple path of least resistance.
Apple has effectively given up support for personal network backups which is why you find it so difficult.
It was easy when it was supported. Since Airport routers went out of production in 2018, this day has been coming for a while.