A couple more things how would seasons, sun rise/set, 24 hour light/dark at the poles etc. work on the bi-polar map?
It would have to be completely different than what the wiki says, since that's a completely different map. I suppose you could use the same "spotlight" idea, which is completely ridiculous to begin with for the actual night/day, but that doesn't explain where the sun would be rising and setting. You also have the same size problem, the Sun doesn't change sizes throughout the day.
How could you possibly have a duration of 24 hour day/night at the poles without the Sun orbiting the poles? On that map it can't orbit either pole because when the Sun would be on the other side the rest of the world will be in darkness. At least the other map works with the north pole in that regard, but not the south pole.
For the seasons I can't think of any explanation other than the Sun moves its orbit from north to south, but then during certain times the orbit would be directly over the poles, among many other problems with that. I really don't see how seasons could work at all with that map. Also what is the mechanism that would cause the Suns orbit to change so drastically for either map?