Kiltie.
Concur completely with your last sentence. However
Different jurisdictions have different rules. In my neck of the woods
the AIP EN-RT 5.7.1 (j) requires us on approach to report " When the ground or water becomes continually visible and flight by instruments is no longer required." (Honoured more in the breach than in compliance, but that's another story.)
No specific phraseology is given, though "Barnburner XYZ, visual" is common. I don't know that this particularly helps the Tower folks, though. Do they need "Request visual approach" in this case, or does "visual" allow them to free up some missed approach airspace?