Hi Roff,
In the left hand drawing it does not matter where the collector resistor appears because all the analysis is based on current measurements. All the currents are given therefore it works just as indicated. But your observation is still interesting to me because it doesnt *look* exactly like the right hand drawing. In other words, short out that resistor and see the same results.
You are also very correct about the official name for the Ie/Ib quantity, but it is still the gain we find when we trace from the base of the first transistor to the emitter of the second transistor. Your observation is still interesting i think though because it's not normally called 'beta'. However, the diagram is clear on what the gain is.
The Btotalcollector is actually B1*B2+B2, not B1*B2+B2+B1 as you have stated. If you look at the current measurements, you'll see that the output collector current is 110ma, not 120ma. Lets do the math to prove this...
First, the input is 1ma, the gain is 10 so the first collector current is 10ma and the sum is 11ma so we have 11ma at the emitter of the first transistor. That feeds into the second transistor base, and that transistor has a gain of 10 so the collector current is 11ma*10=110ma, and that is clear on the diagram as well. Now lets it with the gains both ways.
Method 1: B1*B2+B2
10*10+10=110, 110*1ma=110ma, exactly as shown in the diagram.