Difference between public, max-age and s-maxage

TLDR:

What is the difference between the Public, max-age=<VALUE> andmaxage=<VALUE>, s-maxage=<VALUE> cache-control syntax?

Question:

For one of my projects, I am looking to reduce the server load via the Cache-control http-header. This project is hosted as a serverless function on Vercel and calls GitHub's GraphQL API on the backend to retrieve GitHub user information. Both the Vercel API and GitHub API are rate limited, so I am looking for the best header to prevent these limits from being hit. To achieve this, I am currently using the following Cache-control header:

public, max-age=14400, stale-while-revalidate=86400

According to the Mozilla documentation, this header should keep both the private browser cache and server cache fresh for 4 hours, while the stale cache can be reused for 1 day while it revalidates on the server. Furthermore, since I am not using an authentication header, I should be able even to remove the Public keyword.

The Vercel cache documentation, however, recommends the following header:

maxage=0, s-maxage=14400, stale-while-revalidate=86400

Based on the Vercel documentation and this Stack Overflow question, I, therefore, think the following header is best suited for reducing both the Vercel and GitHub load:

maxage=14400, s-maxage=7200, stale-while-revalidate=86400

To my understanding, with this header, the cache will be fresh for 4 hours for individual users while the Vercel server refreshes the cache every 2 hours, and a stale cache can be reused for 1 day while it revalidates on the server.

As I am uncertain about the difference between the Public, max-age=<VALUE> and maxage=<VALUE>, s-maxage=<VALUE> syntax, I quickly wanted to double-check my understanding.

According to the Mozilla documentation, these two syntaxes should result in the same behaviour if the <VALUE> is equal between maxage and s-maxage properties. However, the Symfony documentation states that the s-maxage flag prohibits a cache from using a stale response in stale-if-error scenarios. My question is, therefore: What is the exact difference between these two syntaxes, and which one would you recommend for reducing the load on both the Vercel and GitHub api?