It is interesting that an 8 or 10 bits ADC is not really 8,10 bits or whatever. In fact, their effective resolution is something like 7.8, 9.5 bits. It is exactly because of the fact the LSB of some samples are chosen randomly. In other words it is supposed that all LSBs chosen based on comparison with the associated reference, however due to the circuit imperfections it is chose randomly (and hence wrongly) sometimes.
A one bit ADC like the one you mentioned, compares the analog value to a threshold voltage (you chose 0.5), if produces the corresponding digital value.
simulate this circuit – Schematic created using CircuitLab
Now, it is supposed that the comparator always works well and decides on the analog input sharply. However, due to the problems in the comparator itself (like kick back noise, thermal noise, meta stability,etc), it does not function as it is assumed.
In regular ADCs with more than one bit, LSB is usually determined as the way you mentioned (comparaing to a threshold) but as one bit case, there are errors that sometime cause the LSB to be random.