Consider updating your vocabulary because the term is often used for both.
Note that rendering to the “screen” really means writing bits at a memory range, which is just one interface for displaying things. Html is another, higher level interface these days.
"rendering engine" has a pretty clear meaning and is a pretty poor term to use for a system for tranforming one kind of text bits into another.
Perhaps you should consider using less confusing terminology in your vocabulary?
https://developer.mozilla.org/en-US/docs/Glossary/Engine/Ren...
Edit: You say "often used for both" but I am struggling to find any other examples. Yhe closest I can find is this extremely poorly named static site generator project: https://github.com/render-engine/render-engine?tab=readme-ov...
Edit2: Man, the appropriation of the term "rendering" by JS people has led to some pretty stupid stuff, like this statement: "SSR, short for Server-Side Rendering, is a technique in web development where the webpage's content is rendered on the server instead of the client's browser."