I've always felt what your showing is a very limited and bastardized explanation (and IMO totally wrong) write up. In fact I think it shows exactly what forced shaw to write his book. This is the type of drivel that led to the total degradation of our dog fighting capabilities. While figure 2 is certainly a classic 2 circle opener off of an even merge lets look at figure one. We have an absolutely poor merge with minimal efficiency "countered" by a tweener merge that creates a fleeting snapshot with an impossible AOT and almost no chance to continue the fight without an overshoot. IMO the last thing the aggressor wants to be is in plane, ideally I'd guess he'd be in a vertical lag roll looking to secure a suitable AOT. The entire concept of the "one circle fight" is to obtain an AOT that allows for a tracking guns solution. A 2 circle fight is any fight that keeps an opponent at bay by forcing a nose to nose interaction. Figure 2 shows the classic example...both planes arrive back in a nose to nose configuration....which is identical to figure 1. The true defining indicator is lift vector orientation not geometry.
By the way, the manual I quoted is the 2007 edition of the US Navy T-45 ACM manual and the diagrams are from the US Navy Instructor Lecture Guide. In other words they are current US Navy doctrine influenced by Shaw and Boyd.
I would have posted my own material that I have taught for over a decade but I thought you would accept the US Navy a little more readily. Murdr has now posted identical information to mine so maybe that will be the end of the confusion.
Flow or fight geometry is really only relevant in a high aspect angle fight. Once the angle has been reduced (i.e. you have gained an offensive position or the bandit has) the fight either results in a snapshot or the offensive fighter is using applied pursuit curves to arrive and remain in the control position or saddle. The fight could quickly degenerate into high aspect again and often does.
But for clarity of discussion, nose to nose maneuvering in the horizontal or vertical plane is a one circle fight favoring the fighter achieving the smallest turn radius. Nose to tail maneuvering favors turn rate. The two situations require different actions.
And there can be multiple swaps of fight geometry (or flow as the Navy calls it) in a very short period of time.
In a 1 v 1 similar aircraft fight it is rare to see two circle geometry unless one of the pilots is pretty savvy.
In a dissimilar aircraft fight, the smart pilot will recognize pre-merge which geometry favors him and try to steer the fight into that geometry.
And to avoid confusion (The intent of all my posts in this thread, it isn't personal), Figure two in my post is a ONE CIRCLE merge (supported by Murdr's post (In case you don't believe the USN)(excellent graphic BTW Murdr).
Its clearly marked "one circle fight"