Understanding the Definition of Most Significant Bits in the Hidden Number Problem
Imagine you are working modulo the prime $p = 2^{255}+95$. Knowing the most significant bit, in the usual sense, of values modulo $p$ gives you almost no useful information—it's virtually always $0$. Knowing the $k$ most significant bits, in the usual sense, gives you roughly $k-1$ bits of information.

For the results to make sense over arbitrary prime fields, one thus needs a definition of most significant bits that is not affected by such quirks, and that's there the weirder definition of $\mathrm{MSB}_k$ comes in, which might be better understood as "suppose you know $k$ bits of information about $x$".