WP Function does not trigger on Webhook API Call

From what I see is the issue within your code.
Your issue is that you try to fetch the post type in your nested if-statements from the current post, even though at the first if statement, you validate against it and break the function.

What you want to do instead is to validate the post type against the $client_id variable.

$client_parent = get_post( $client_id );
$post_type="";
if( ! empty( $client_parent ) && isset( $client_parent->post_type ) ){
    $post_type = $client_parent->post_type;
    if( $post_type === 'note' ){
        //run your logic
    }
}

In case it’s still not working, you might want to take a look at the documentation of the update_post webhook action.