I rechecked using the UK site at godaddy -- godaddy.com/en-gb

and couldn't reproduce it. The steps were:

1. pick a novel domain 2. register and account with name and throwaway email 3. click checkout 4. empty the cart 5. enter a new novel domain name but don't Enter or Search 6. wait two days -- observe no email 7. click search 8. wait two days -- observe no email.

So I conclude they don't do the keystroke logging any more, or at least not on the UK site.