The word "Sushi" refers to ingredients plus Sushi Rice, which is rice mixed with Sushi Vinegar (rice vinegar, sugar and salt).
When rice is mixed with Sushi vinegar, it is called "Su-meshi."
The word "Su" means vinegar, and "Meshi" means rice.
It is said that people started to abbreviate the word "Su-meshi" to "Sushi," which became the name of the dish we now call Sushi.
Therefore, as long as the dish uses Sushi Rice it is called Sushi.
The ingredients to mix with Sushi rice can be anything - vegetables, fish, and meat, both raw and cooked.
The style of Sushi as we know now, is said to be invented by Hanae Yohei circa 1830 in Tokyo. For his Nigiri Sushi, Yohei used ingredients (mostly fish) caught in Edo Bay, thus the name Edo Mae Sushi (Edo Style).
Because of the lack of refrigeration, Yohei used such techniques as marinating fish in soy sauce (zuke), curing and cooking. Anago/Sea Eel, and Clam were some of the cooked ingredients.
So you cannot talk about authentic Sushi without talking about cooked fish.