Von Neumann Extractor - Which bit is retained?

Question

Which bit is retained in the Von Neumann debiasing algorithm? 00 and 11 are discarded and 10, 01 are retained but is the first or the second bit retained or does it matter?

In other words:

first: 10 -> 1 , 01 -> 0

second: 10 -> 0 , 01 -> 1

Original Paper (appears to be first but could be interpreted either way):

https://dornsifecms.usc.edu/assets/sites/520/docs/VonNeumann-ams12p36-38.pdf

Examples (first digit accept):

https://en.wikipedia.org/wiki/Hardware_random_number_generator#Software_whitening

https://en.wikipedia.org/wiki/Randomness_extractor

http://pit-claudel.fr/clement/blog/generating-uniformly-random-data-from-skewed-input-biased-coins-loaded-dice-skew-correction-and-the-von-neumann-extractor/#more-410

https://people.seas.harvard.edu/~salil/pseudorandomness/extractors.pdf

Examples (second digit accept):

https://www.researchgate.net/publication/38359648_Iterating_Von_Neumann's_Procedure_for_Extracting_Random_Bits

https://www.esat.kuleuven.be/cosic/publications/article-2628.pdf

Examples of the second are present in papers regarding an Iterated Von Neumann algorithm.

Answer 1

It doesn't matter at all, since both $01$ and $10$ have the same probability $p(1-p).$