Correct. They are 2 separate things. Coffins are still popular in the UK and throughout Europe – although the sides were made from a single board they were kerfed on the insides at the elbow to allow the board to be bent to achieve roughly the same shape. The Western style with it’s angled sides were considered a bit crude by the English undertakers. The coffin fell out of style in favor of the casket because people did not want to be reminded of what was inside by the very shape of the coffin itself. Also, in the UK they don’t really use handles as the coffins are carried out on the shoulders.

Good video from youtube explaining more on the history.

https://www.youtube.com/watch?v=nUaqKZnClGQ