The USNA application "check box" is private pilot. It is asking for the type of pilot certificate you have, not the class or type.
"Private Pilot" is the certificate. There are 6 types of certificates. Student, Sport, Recreational, Private, Commercial, Flight Instructor and Airline Transport Pilot.
You can have a "Private Pilot" license for different categories of aircraft. Different categories would be Airplane, Rotorcraft, Glider, Lighter than Air, etc. Within each category there are different ratings Airplane would be Single Engine Land (typical starting point), Single Engine Sea, Multi Engine Land, and Multi Engine Sea. Rotorcraft ratings are Helicopter and Gyro Craft. And then for each rating there are additional endorsements and ratings. You can be qualified for multiple categories, ratings and endorsements, depending on training, testing and of course money.
Start saving your money. Helicopters rent for $400-$600 per hour. The minimum flight time for a private pilot license is 40 hours, but typically 50-60 hours are required. Do the math, $16,000 to $36,000!! Significantly more than a SEL Certificate which would be $5,000 to $10,000. If you really want to be a private pilot, do it in an airplane then add helicopter training as funds are available. Or even better let the Navy train you and you will get paid.