In FAA speak you are a commercial pilot with your ATP written(s) passed. In order to get a FAA commercial the 2003 post is still valid but he left out the requirement to hold a FAA private certificate to be eligible for the commercial. You can do the 61.75 paper shuffle and get a FAA private based on your commercial. Also TSA started their background check requirement in 2004. Since no training is required for the 61.75 you can get that without the background check but you would need it before you started training in the aircraft.
If you have the hours required for the ATP you can skip the FAA private and commercial but you must complete an ATP Certificate Training Program before you take the written test if you want a ME ATP.