You're simply wrong here. The algorithm is O(log n), whether you agree to it or not.
First of all, there's absolutely nothing that says that we have to assume that the number of lines is less than 2^64. Big O notation does not care what happens below that number, it only cares what happens when you go to infinity, which is a lot larger than 2^64. At that point you need arbitrarily sized integers, which are not fixed-space. Hence O(log n). Word size or RAM model does not matter one whit.
This is how Big O notation works, it has nothing to do with practicality. As another example of a similar thing, consider Sorting Networks. The AKS sorting networks has the best "asymptotic depth" of any known sorting networks (O(n log n) depth, i think), but the hidden constants are so massive that there are no situations where the AKS networks are actually practical, for any reasonable input they are always beaten by other (asymptotically worse) networks. That doesn't mean that they "aren't O(n log n)" or whatever.
Second: if we were to actually write this algorithm in the way which is most conservative with space, we wouldn't use a 64-bit integer in the first place. If we only care about memory, we'd start the count with a single-byte datatype (i.e. unsigned char) to keep the count. The algorithm now uses 1 byte of space. When that overflows (i.e. reaches 256 lines), we would change to a bigger data type, i.e. unsigned short. The algorithm now takes 2 bytes of data.
When THAT overflows, we again promote the datatype to an unsigned int. Now we use 4 bytes of data. When that overflows, we finally promote it to a uint64_t. Now we're on 8 bytes of data. When that overflows... (etc.)
See how the memory usage is growing? And how it's growing exactly with the logarithm of the number of lines?
You're just wrong about this. Yes, you can absolutely say "assuming storage of all numbers is O(1), then the algorithm is O(1)" (essentially what OP said) but that is not a default assumption and it's not accurate to reality (since actually storing numbers requires O(log n) of space). And yes, branch prediction misses and cache misses are far more important factors here, but that doesn't change the fact that it uses O(log n) space. That's just simply true.
Also: we're not dying on a hill. OP made an entirely accurate and true statement, which was met with an avalanche of people criticizing him using incorrect arguments and misunderstandings of Big O notation. We're just saying that he was correct all along.