So you mean they are gone when you log out or refresh?
Normally that can't be ...
Please check the contents of Table administrator if it's there.

The note is in the table ok, but it doesn't show in the Global Notes box in the Admin CP.

What exactly is in the table and what does it show (or not)?
And which mode do you have activated?

I have Only Global Admin Notes enabled. The note I entered is in the Administrator Table but that note does not show up in the box.

I ask again:
What exactly is in the administrator table (fieldname, value)?

Does this help? I am userid 2, by the way.

Yes. It clearly shows that there are no global notes at all.
Edit: OK, there was a wrong condition. Should be fixed now.
Guess I should not only have tested display but also saving :D

Much better. The note is showing up now on refresh. ;)

Nice to see you taking this one step further Kirby.. :)

I think he stumbled on that step, though, but recovered nicely. ;)