The help page linked above documents the variables we POST to Common/Product Notification URLs. There's no way to restrict which variables are sent, but typically you would write your listener script to parse for just the variables you need and ignore the rest. Note that a few of those variables are only sent to a Common URL, and a few others only to a Product URL, and not all variables listed there are sent every time, depending on the nature of the order (e.g., shipping address vars may be omitted for an order that didn't require shipping).
The URL query parameters we append to a Common Thank-you Page URL or Redirection URL are unrelated to POSTing order data to a Notification URL. This is really only useful for dynamically customizing how your page is generated, if and when the buyer arrives there; it's not really a reliable means of transmitting order data to your end, as any given buyer may not access the redirection to your URL for various reasons. This help page documents the URL query parameters we append when redirecting buyers to a URL at your end:
