Update custom settings field in plugin

Your code is saving the option, but your input field is not actually displaying the option’s value because you’ve used the wrong option name, which is url_field_name and should instead be webhook_url_field_name (the second parameter value for register_setting()). So the correct HTML would be:

<!-- I wrapped the value attribute for brevity -->
<input type="url" class="webhook-url" id="webhook_url_field_id" name="webhook_url_field_name" size="100" placeholder="e.g. https://hook.integromat.com/j2bb812345678"
    value="<?php echo get_option('webhook_url_field_name'); ?>">

And remember to escape the value, e.g. using esc_attr(), i.e. echo esc_attr( get_option( 'webhook_url_field_name' ) ).