Which one of these two sequences is random, and which one is not?
The second sequence is not random.  Let $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ be random, iid Bernoulli $1/2$ random variables.  Let $\beta = B_4(\alpha_1 \alpha_2 \alpha_3 \alpha_4)$.

What is the distribution of the random variable $\beta$?  Answer: $\beta=1$ if at least two of the $\alpha$'s are $1$, so $\Pr[\beta=1] = 11/16$.

In other words, $\beta$ is biased towards $1$.  It follows that the second sequence is not algorithmically random: it is a set of independent Bernoulli random variables with $p=11/16$, i.e., the outcome of an infinite sequence of tosses of a biased coin.