I leave the separator as default of "\n\n".
Which means when there is an empty line after my first paragraph (or image), then the excerpt ends.
I haven’t had to override it.
You can use something like the more bit. Or a span. But only if you artificially wanted an excerpt to say be only a portion of the first paragraph (its simpler just to make that it’s own line and keep the default).
Or if you want to stretch your excerpt to two paragraphs, which is not a good idea for length.
The advice above for setting a config level override is good.
You might need to quote the value to avoid any funny YAML evaluation
excerpt_separator: "<!--more-->"
Another idea is you can set up a description text field on the frontmatter. Which might be the first sentence of your body content or maybe you write a summary instead.
Then use that instead of an ex excerpt.
---
title: Bike riding on sunny days
description: This sums up my post, about fun in the sun.
---
Then use a for loop going over posts.
{{ p.title }}
{{ p.description }}
I would not call the field “excerpt” in case it breaks things but summary or description are apt.