Correct, they just see that you have an account with someone. Unless the account belongs to them, they will not see who the account is with.
Usually that statement accompanies a default entry. It's removed 6 years from the default date.
I found 2 entries, one for the account which showed all green, and marked as Settled. No statement on this. In fact some still remained giving a positive effect on my credit file at the end as it showed a positive history
Then a second entry for the default, which then showed as Partially Satisfied with the statement as you say "Payment received as a full and final settlement although the payment would not fully clear the balance." or similar wording, cannot remember, against the default entry.
An account that is not shown in default, will remain on file for 6 years from the settlement date, so something to be concerned about if it's shown on a settled account and not the default entry.
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