I think the answer depends on which context we are talking about here (oh man, engineers like using terms differently for different contexts).
Context 1: if you are talking to a friend who just knows basic knowledge about the web...
The browser engine in this context refers to a software that powers your browser and responsible for displaying content on screen. If you search for browser engine in Wikipedia, it tells you popular browser engines include Webkit, Gecko, Trident, and so forth (https://en.wikipedia.org/wiki/Browser_engine).
The rendering engine in this context would be...if your friend knows this word, he or she should fall into context 2 :)
Context 2: if you are talking to a friend who knows how browsers work and all the crazy magic behind them...
The browser engine in this context refers to the browser process which is mainly responsible for managing all render processes and displaying UI. So in your question, you mentioned about
The browser engine: marshals actions between the UI and the rendering
engine.
This is correct as well. If you look at the Chromium's architecture, you will notice that the browser process/engine coordinates page content with rendering processes.
The rendering engine in this context refers to a program that constructs DOM, executes JavaScript, and layout out web pages, e.g. Webkit, Gecko, Trident. A rendering engine consists of two primary components: WebCore which contains core layout functionality and JavaScriptCore where JavaScript interpreter V8 lives.
Your friend seems to be an expert and must also know about the rendering process, which is responsible for constructing a web page. A rendering engine is just a crucial part in the rendering process.
The following image shows the high level architectural overview of Chromium architecture (Google Chrome open source version). If you want to read more about the magic behind modern browsers, you can check out this post: https://medium.com/@zicodeng/explore-the-magic-behind-google-chrome-c3563dbd2739
