Where to hook my plugin’s action

Something important – when you create a new post the first thing WordPress does is an auto-save, so yes the first time the values should be empty. Perhaps that is what you are experiencing – I don’t see your full code, but I do see

The plugin for the custom meta data is hooked in the “save_post”
action.

Also, while editing WordPress also does an auto-save every 60 seconds by default. So perhaps check for auto-saves in your code, or you could try disabling auto-save completely.

function disable_autosave() {
    wp_deregister_script('autosave');
}
add_action('wp_print_scripts','disable_autosave');