How to cache openstreetmap tiles

I wanted to build an openstreetmap tile cache server to lower the impact of my application in openstreetmap servers. A motivation for this is a moral one and also the limits they have to request their resources.

I have an nginx server with the following configuration in nginx.conf

proxy_cache_path  /tmp/cache levels=1:2 keys_zone=openstreetmap-backend-cache:8$
proxy_temp_path   /tmp/cache/tmp;

upstream openstreetmap_backend {
  server  a.tile.openstreetmap.org;
  server  b.tile.openstreetmap.org;
  server  c.tile.openstreetmap.org;
}

Note that the directory /tmp/cache must exist and be writable by the nginx process user, usually www-data

And the following configuration in the server block

location /osm_proxy/ {
  proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
  proxy_set_header X_FORWARDED_PROTO http;
  proxy_set_header Host $http_host;
  proxy_set_header Host "tile.openstreetmap.org";
  proxy_set_header User-Agent "my-proxy.example.com/0.01";
  proxy_cache openstreetmap-backend-cache;
  proxy_cache_valid  200 302  365d;
  proxy_cache_valid  404      1m;
  proxy_redirect off;
  if (!-f $request_filename) {
    rewrite ^/osm_proxy(/.*)$ $1 break;
    proxy_pass http://openstreetmap_backend;
    break;
  }
}

To use this server from leaflet, I configure leaflet tile layer like this

L.tileLayer('https://my.tile-cache-server.com/osm_proxy/{z}/{x}/{y}.png')

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How to cache openstreetmap tiles