1s are 1s, 0s are 0s, right?

You answered it yourself. Everything makes a difference.

During last decade or two there were those who worried about 1's and o's while some of us were happily listening to vinyl, it does not even have the error correction system (adding 1's and 0's on its own)

Someone said "There are 10 kinds of people in this world, those who know binary and those who don't"

Happy listening...and hope you don't miss any beat!

Unsound...The electrical signal that transmits digital data (a pulse train) can be greatly distorted, and the receiver can still reconstruct the original data perfectly,

Your example: 010101 vs 0 1 0 1 0 1. I believe you are thinking of jitter.

How about 16 bits: 01 0 10 1 0 1 01 01 010 1.

If the receiver outputs 16 bit words to its D/A at uniform intervals based on its own clock, it doesn't matter how irregular the incoming serial data transmission was.

Furthermore, some of the zeros can be misread as ones, and the correct original word can still be recognized, thanks to the error correcting encoding of the data, but that's another topic.

Vinyl has its own set of problems.

I'm not so sure about the common explanations for sonic differences caused by cables («impedance») and transports («jitter»). My own experiences with digital cables tell me that impedance isn't the decisive factor, there are others, like materials used and conductor geometry/profile. I found the sonic impact of transports to be even clearly higher than with cables. Given that many good transports have very similar jitter patterns on a low level, I don't think that jitter should be seen as solely responsible for the sonic differences, which according to my experience often appear in the level of the sound balance -- which seems to be impossible anyway, looking at the even frequency responses measured with DACs, independent of the connected transports.

Believe me, the differences are real, not imaginated, and very obvious in the case of transports. Well, the mentioned explanation attempts obviously are the only ones available at present, so it's logical to give them a certain plausibility. But this shouldn't be mixed up with assured knowledge.