[tgb]

Won't their definition conflict with the rectilinear definition? Specifically, there are distinct rectilinear configurations which are homotopic. With two bricks connected by at corner, there are two rectilinear configurations but both are homotopic. So their method used to count "homotopy equivalence classes which contain at least one rectilinear configuration" would yield a lower number than the previous result. This seems to contradict when the paper says that their definition extends the previous one.

[teraflop]

Where are you getting two from? By the definitions on this page, it seems to me like there are four corner-connected rectilinear configurations of two bricks, and also four homotopy equivalence classes, so there's no inconsistency.

    +--+       +--+
    |  |       |  |
    |  |       |  |
    |  |-+   +-|  |
    +--+ |   | +--+
      |  |   |  |
      |  |   |  |
      +--+   +--+
    
    +-----+       +-----+
    |     |-+   +-|     |
    +-----+ |   | +-----+
         |  |   |  |
         |  |   |  |
      	 +--+   +--+

[tgb]

Your ascii art has convinced me! I imagined you could rotate the left two into each other but that makes no sense.