I think the problem lies in turn vs. round.
ZetaStriker is describing a round, while PeelSeel2 is describing a turn, in a 3e kinda way, I'm pretty sure. 3e isn't my cup of tea, esp. at this time of the morning.
The
round begins with the highest initiative number and proceeds to the lowest, with your turn somewhere in there amid the NPCs and other PCs.
Your
turn begins when your initiative number comes up in a round, and lasts from when you begin your upkeep
ie, ongoing damage, regeneration, "other effects" as mentioned on page 268, to actions on your turn, to the end of your turn, on page 269, where you make saving throws, etc. (It's not AFAIK from the start of your initiative to the start of your next initiative. At least, not in 4e.)
Good example of this terminology in action are the 2 kinds of Triggered Actions, OAs use the term "turn" while Immediate Actions use the term "round" since an OA is done on a specific creatures turn, while an interrupt or a reaction can go off at any point when the trigger is met.