The local test site is now working perfectly, thanks. So it's not a K2 issue.
I also downloaded and installed the complete site onto my local computer (I'm using XAMP) - and the K2 settings work fine!
So the issue is only with the live site. I've reinstalled Joomla on the live site as suggested, but it made no difference.

I don't know if any of this helps.

RESOLVED
It was in fact a server issue. The service provider has fixed it.

Thanks Fotis,
I tried to send more info before that email but it got lost on Send, so I then cut it short as it was late in the night.

It's true, it has gone on and on and on; and I thank you for persevering with me. A lot of time was wasted chasing this down. The service provider fixed it in about 10 mins.

Here is their correspondence:
It seems your authorisation may have been revoked by Mod Security, which would explain the 403.
I have whitelisted any blocks that may have occurred by the IP address.
It's probably worth mentioning that whitelisting can only done by us.

And then this for clarity:
Authorisation getting revoked means that there were some temporary blocks applied to your IP for hitting ModSec rules often.

It seems K2 does things that some security programs don't like. You might like to look into this one day.

Thank you once again. If others were following this thread, I hope they learned something.

Regards, Nick