19x19 diagram  

A correct statement of the no-suicide rule would be this: it is illegal to place a stone on the board that would not capture any enemy stones and whose block would have no liberties.

A more serious problem is with Theorem 2, which is outright false. Here's my counter-example:

19x19 diagram  

Clearly there is only one black block b. Also, b is safe, because if black passes every single turn, then white will be unable to fill either of the regions f and g. So the set of all safe black blocks is X = {b}. However, contrary to what the theorem states, X is not unconditionally alive. By definition, in order to be unconditionally alive, X would need to have two distinct vital regions. Then, by definition of vitality, b would have two distinct healthy regions. The only possible healthy regions would be f, g and h, since no other region is surrounded by black stones. But neither g nor h is healthy, because they don't satisfy condition (3) of the definition of small black-enclosed region: they contain empty intersections in their interior! So b doesn't have two distinct healthy regions, which is a contradiction, thus proving that X is not unconditionally alive.

My analysis above suggests that condition (3) is the problem. In fact, intuitively, I have no idea where it comes from. Why do you need the interiors of the regions to be white, anyway? Also, the health condition that "every intersection in region R is a liberty of block b" implies condition (3), so it is also nonsense. Why would need every intersection in the region to touch the block? Intuitively, all you need is for each block to be touched by at least one region, so that it becomes protected.