Confusing terminology about stream ciphers
If you used the randomly generated keystream bit $k[i]$ twice (think of it as a random variable, uniformly and independently generated at time $i$) at positions $i,i+\tau,$ then you'd have
$$
c[i]=p[i] \oplus k[i]
$$
and
$$
c[i+\tau]=p[i+\tau] \oplus k[i],
$$
then adding the ciphertext bits would yield the plaintext bit "difference"
$$
p[i+\tau]\oplus p[i],
$$
destroying the property that
$$
c[i]\oplus c[i+\tau],
$$
is independent and uniformly distributed, and making the conditional distribution of $c[i+\tau]$ given $c[i]$ dependent on the plaintext difference $p[i+\tau]\oplus p[i],$ which need not be uniform.

Thus the ciphertext sequence is no longer purely random (independent and uniformly distributed) which destroys the perfect secrecy property of the OTP system, since the ciphertext distribution is no longer independent of the plaintext distribution.