The 162 comes from unique full trips. The trips can be very similar but a single different location makes it a different trip. Examples are Knight Academy - Bridge - Bazaar - Small Island - Knight Academy and Knight Academy - Bridge - Bazaar - Small Island - Light Tower. This is also assuming that the transition at the very beginning of the stage is different from the last spot to first spot transition. It could very well be the same in which case the number would be smaller. I've had plans to try and record every spot with hazards with my crappy capture card but haven't gotten around to it.
Problem: The second path you describe is impossible. The stage goes through exactly four stops before returning to the start position, and each stop selects from a specific and unique set of locations, with no overlap.
1. Knight Academy, Light Tower, or Windmills
2. Bridge, Plaza, or Residential District
3. Bazaar, Isle of the Goddess, or Statue of the Goddess
4. Small Island or Waterfall Island
3 * 3 * 3 * 2 = 54 possible paths the stage can take.