1. The introductory passage of page 7 states "During a normal round..." That means there are exceptions.
2. The definition of a round, as given in the PHB is like that of "a week" - it can either mean "Sunday to Saturday", or it can mean "from now until next Friday". So, a round can mean "from the top of the Init order to the bottom", or it can mean "from my turn until my next turn".
The descriptions given for Swift and Immediate actions, then, clarify how #1 and #2 work out in practice. In particular:
#3. "If you use an immediate action when it isn't your turn, you can't use another immediate action or a swift action until after your next turn."
and
#4. It does not say that using a Swift action prevents you from using an Immediate action immediately after your turn ends.
So that's it. In the normal case, either:
A. On your turn you'll use a Standard, Move, and Swift action (or some other allowed combination)
OR
b. You'll use an Immediate action when it's not your turn, and then on your turn you'll use a Standard and Move action (or a Full-round action).
But there is a corner case where you use your full set of actions on your turn (per A) and then before the end of that round you find the need to use an Immediate action. And that's allowed - that just means you've used up your Swift action for your next turn. (It's allowed because of #3 and #4, and it fits with #1 because that means it's no longer a normal round.)
Here's the problem with your argument.
If you are going to use the example of the round being 'but it usually means a span of time from one round to the same initiative count in the next round'. There is a huge problem with that. It's stated pretty clear here: 'During a normal round, you can perform a standard action and a move action, or you can perform a full-round action. You can also perform an immediate action or a swift action, and as many free actions as your DM allows.'
So, if you were to perform a swift action during your turn, the 'round' isn't up until your next turn. So, you couldn't do an immediate action.