= Solution
Before collecting any observations, a uniform success-chance <prior distribution> gives
$$
\Pr(Y=y)=\binom ny\int_0^1\theta^y(1-\theta)^{n-y}\,d\theta
=\binom ny\mathrm B(y+1,n-y+1)
=\boxed{\frac1{n+1}},\qquad y=0,\ldots,n.
$$
Thus the <uniform-count predictive property> assigns equal probability to every possible number of successes in a fixed-size batch under the <prior predictive distribution>. It is uniform over counts, not over all ordered binary sequences. After observing $x$ successes in $m$ trials, its familiar one-step <posterior predictive probability> is $(x+1)/(m+2)$; the count-uniform property refers to prediction before those data are observed.
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