How does the Innistrad "no spells / two or more spells cast last turn" transform trigger work?
If it's your upkeep, the last turn is the turn that has most recently concluded (usually your opponents). If it's your opponent upkeep, the last turn is usually yours.

What is a turn?


  500.1. A turn consists of five phases, in this order: beginning, precombat main, combat, postcombat main, and ending. Each of these
  phases takes place every turn, even if nothing happens during the
  phase. The beginning, combat, and ending phases are further broken
  down into steps, which proceed in order.


So a turn contains all the phases. But when is the upkeep?


  501.1. The beginning phase consists of three steps, in this order: untap, upkeep, and draw.


It is in the beginning phase. The whole turn before this point is the turn of the previous player.


  603.2e If a triggered ability’s trigger condition is met, but the object with that triggered ability is at no time visible to all
  players, the ability does not trigger.


That whole turn will be checked to see if the condition has been met to so it can trigger.