According to Wikipedia, binary search concerns the search in an array
of sorted values.
The more general concept of divide and conquer search by repeatedly
spliting the search space is called dichotomic search (literally:
"that cuts in two"). The use of dichotomy may be considered in other
contexts, as sonn as you have something to split. It is actually the first expression I learned (in high school, I think, and that was long ago), including in cases where you might want to call it binary.
Afaik, "dichotomic" does not imply that the two parts are (nearly)
I do not know that binary is reserved to search in a space of size
Dichotomic is clearly the more general term, but it may sound pedantic
to some who might instead improperly use binary.
Your example (1) is strangely stated, as one does not consciously ask
for binary digits, but rather for comparison with the median of an
interval. But it could qualify as binary.
Youe example (2) is unclear. Just splitting in two should be called
dichotomic. Now, as you seem to hypothesize (strangely) a way of
making 2 equal parts, I am not sure.
But a guessing game, where people ask questions that are answered by
yes or no is clearly dichotomic.
My own guess, no reference given:
The original expression was probably "dichotomic", but with the popularity of binary systems, binary computer, etc., the term "binary" became more popular.
One other factor that may have played an important role is that binary search (as well as dichotomic) is based on binary choices. Now the expression "dichotomous choice" does exists, but is much less used than "binary choice", which appear about 6 times more often on the web.
So this may have influenced that. We should remember that though we are largely immersed in binary number (I mean we, computer scientist), most people are not
and are nor concerned with binary numbers, but will easily talk of a binary choice. It is true that binary search is a topic for computer scientist, but short of a reliable reference to the contrary I will not believe it comes from binary numbers in any direct way.