I need help understanding trailing slash behaviour in Nginx

I’m setting up nginx as a reverse proxy for squaremap (a world map viewer for Minecraft servers) and encountering unexpected behavior with trailing slashes. I’ve followed the squaremap documentation for serving with nginx acting as a reverse proxy (https://github.com/jpenilla/squaremap/wiki/Internal-vs-External-Web-Server), but I’m confused by the results. Here’s what I’ve tried:

squaremap is running at 127.0.0.1:39000

Configuration:

location /squaremap {
   proxy_pass http://127.0.0.1:39000;
}

Result: Accessing https://example.com/squaremap returns a 404 error.

location /squaremap {
    proxy_pass http://127.0.0.1:39000/;
}

Result: https://example.com/squaremap shows a blank page, but https://example.com/squaremap/ works fine.

location /squaremap/ {
    proxy_pass http://127.0.0.1:39000/;
}

Result:

https://example.com/squaremap redirects to https://example.com/squaremap/ and then displays the web interface. https://example.com/squaremap/ works as expected.

In my attempt to figure out what was happening, I read part of the nginx documentation on proxy_pass. However, I’m not sure if my interpretation is correct. My understanding is:

1. If there’s no URI in the proxy_pass directive, the request URI is passed to the upstream unchanged.
2. If there is a URI in the proxy_pass directive, the part of the request matching the location directive is substituted by the value of the URI in the proxy_pass directive.

Based on this, I created a table of what I think is happening in each of the above cases:

My questions are:

1. Is my interpretation of how nginx processes these requests correct?
2. Why do I get different results in cases 2a and 3, even though they seem to send the same request to the upstream?
3. Why doesn’t it work without the trailing slash, but works with it?
4. Is there a configuration that would allow both /squaremap and /squaremap/ to work correctly without a redirect?

I’d appreciate any insights into understanding this behavior and how to properly configure nginx for this use case.