Working within the three print runs for these first TOA $1 green stamps the second print run is the first to have text in the margins - ANKH-MORPORK - THE YEAR OF THE PRAWN on the left and TEEMER & SPOOLS TOWER OF ART - ONE DOLLAR on the right. The top left and and bottom right stamps also have the BP CE AB logo in the shorter margins.

Is it possible to identify common stamps from each the three print runs? As the same wording on the side selvedge appears on the second and third print runs only the absence of the text on a left or right margined stamp can tell you that a stamp comes from the first run. If there are there different print runs then one would expect slight differences between them in terms of shades, intensity of print, print resolution or even the paper. If the sheets from three print runs were delivered in multiple batches there would be multiple slight differences. Here are some stamps from my collection showing some subtle differences.
There are slight shade differences and intensity of colour, but significantly there are more than three variations, and I have not included all the stamps that I could. The leftmost stamp also appears to have a different gum; unfortunately not noticeable on a scan, so you have to believe me on this. The paper seems identical from the front so it is the gum that differs. The fifth stamp is from the third print run, identified from the selvedge logo, while the first stamp may be from the second or third print run.

The stamps which came from the so-called second print run probably outnumber those from the original and final prints; perhaps even those combined. The demand for the stamps was unexpected, so the original run would have run out soon, and there was have been a move to add some text decoration to the sheets.  The first stamp sheet to include a logo and text was the Assassins Guild Three Pence, in January 2005. As all subsequent sheets followed this pattern it is likely that the second print run of TOA sheets was around the same time. The final print run, from the sheet numbers are known for certain from the numbers of new sports released, only contributed 1 in 6 of the total numbers. The second print run stamps were available for twice as long as the first run, and most likely overlapped the third run sheets, which were only created for additional sports.

Different printing firms were used over the period that this stamp was available, and different printing equipment would have been used both between firms, and over time. Sheets were delivered to the shop in varying amounts; sometimes as few as just five sheets.
At least one batch of sheets made use of the space below the block of 16 TOA stamps by including the Red Nose Patricians there. Those would have been guillotined off before any perforating was done. This would have been a relatively small batch if only 100 of the Patricians were required although more would have been printed than needed in case of perforating problems occurring; just 10 sheets would have been needed, This batch would have been within the second print run period, with the Patricians being required in March 2005. The idea of utilizing the spare space was continued with the third print run to provide new TOA sports.

This imperforate sheet may be an example from the second print run (click for a larger view). Like the third print it has both text down the sides and the CE AB BP icons in the top and bottom margins. What sets it apart from known third prints is that the falling man icon in the margin is reversed to those of the third prints. There is no evidence that extra stamps were included in the space next to the panel of TOAs, but those may have been before it was scanned.