Input distribution in FLP Impossibility Lemma 2 in "Impossibility of Distributed Consensus with One Faulty Process" is as follows:


  LEMMA 2. P has a bivalent initial configuration.


They prove this by showing that the opposite assumption creates a contradiction. As part of that proof, they state:


  Let us call two initial configurations adjacent if they differ only in the initial value x, of a single process p. Any two initial configurations are joined by a chain of initial configurations, each adjacent to the next. Hence, there must exist a 0-valent initial configuration C0 adjacent to a 1-valent initial configuration C1.


In other words, there exists two "adjacent" sets of inputs that differ in only in the value of one of the inputs.

But if I use the output of a totally correct consensus protocol to ensure that the inputs are always either all 0 or all 1, the resulting two sets of inputs would not be "adjacent".

So it appears they implicitly assume that every combination of inputs must be possible. Why can they make this assumption?
The authors' goal is to rule out protocols that work for every possible input. Hence, in their proof by contradiction, they can assume that the protocol works correctly on every input.

The specification of an algorithm should include all constraints on the possible inputs. For example, binary search assumes that the input array is sorted. The specification of consensus protocols doesn't specify any constraints beyond the processes' inputs being binary.