TLDR: On most cases, go for Largest Contenful Paint.
FCP, FMP, LCP? What?
First contentful paint measures the time when the first visual dom element is painted on the canvas. First meaningful paint is the moment when "important" content is painted on the screen. Largest contentful paint measure the time at which the largest element above the fold is rendered.
All of them give an indication of when the browser painted information on the page. The closer it is to what the user wants the see the better the experience we can provide an optimise for.
The main difference between this three metrics is the main element measured to decide the metric.
No. These are very different metrics in the method they use to estimate completion and relevancy. It is known that the bigger element of the page is the one we consider the most important when deciding if a page is loaded or not.
First contentful paint it's useful. On single page applications measures at what moment do you see the loader. You know at which time users have its first feedback that the web app is loading.
On server side rendered apps gives at what time and which is the first thing the users sees. If you have a blog and the main content of the article is the one you intend to show it might be more relavant for the metric to show that instead of a banner/header or whatever the biggest element might be.
FCP, FMP, LCP sound very similar but there is a sutil difference between them. From the first element render to the most prominent one, depending on the anatomy of your page one might suit you better. If in doubt, go for the Largest contentful paint and see if the element and number makes sense to you.