I'd be curious to know just what aircraft had that starting order. I flew the Connies, DC-6/67, and NEVER saw that order. But the initial question related to piston aircraft.
Among others, the PB4Y-2...which we typically started 1, 2, 3, and 4. Sometimes just the inboards, sometimes all four, sometimes the inboards then the outboards, sometimes for variety we'd start from the right and work left, and sometimes from the inboard working out. However, generally we'd start far left and work right, 1, 2, 3, 4. Most others operating alongside us, which included DC4's, DC6's, and DC7's, did the same.
In the C-130, we operated typically 1, 2, 3, and 4, but also frequently started or taxied on just the inboards.
In the B747...1, 2, 3, 4.
In the C97...1, 2, 3, 4.