While I agree with what you've posted here, let me point out that the term in question has been used before.
Dwight D. Eisenhower's vacation home outside Denver (near where his wife Mamie's family lived), Lyndon Johnson's LBJ Ranch, Richard Nixon's place in San Clemente, California, Jerry Ford's Palm Springs and Aspen houses, and Ronald Reagan's ranch outside Santa Barbara were all called "the Western White House."
Franklin Roosevelt's place in Warm Springs, Georgia, John Kennedy's Palm Beach compound, Jimmy Carter's Plains, Georgia house and a Key West house used by both Harry Truman and Eisenhower (one of the few things they agreed on), all due to their Southern locations, were known as the Summer White House.