Like I said - a VERY short timeline for illustrative purposes. But in my defense - just a quick quote from the Encyclopedia Brittanica -
"Destroyer - fast naval vessel that has served a variety of functions since the late 19th century. The term destroyer was first used for the 250-ton vessels built in the 1890s to protect battleships from
torpedo boats. These torpedo-boat destroyers, as they were called, then became super torpedo boats themselves, so that, by
World War I, they were commonly deployed ahead of the battle fleet to scout for
the enemy fleet, beat back its destroyers with cannon fire, and then launch torpedoes against its battleships and cruisers."
I have many other sources that I have read over the years and I believe this is the correct history.
As you stated, the ASW aspect was a much later developement - in essence the original purpose was much as land-based cavalry - screening, raiding and pursuit - the fuctions of a sail-based frigate.
Also - this quote from
http://wapedia.mobi/en/Destroyer :
"The first ships to bear the formal designation "Torpedo boat destroyer" (TBD) were the
Daring class of two ships and
Havock class of two ships of the
Royal Navy, developed in 1892 under the newly appointed
Third Sea Lord Rear Admiral "Jackie" Fisher."
Cheers!