A GNOME Shell extension with an indicator in the panel displaying status of entered web server URLs using emojis:
initializing...
server down/unresponsive/timed out
malformed URL
server up and responding
The panel icon shows the worst status from the set of all server statuses in the popup menu. Server down is considered worse than bad URL.
There's a configurable timeout on the requests (10 second default) and HTTP response codes 2xx and 3xx are treated as success.
There's a per server option to show a notification when it's down. It defaults to false but can easily be switched on.
Servers can be made visible (default) or hidden, where the server is not displayed in the menu and no requests are made to it. Many thanks to xingwangzhe for that enhancement.
Install at extensions.gnome.org or use Extension Manager.
To customize, the four .svg files in the assets folder can be swapped out with your own icons. Just keep the same filenames.
I thought about adding a backup/restore function but I realized DConf Editor already works well. Just copy the JSON text to a backup file and paste to restore from a file.