I asked a more general version of this question in an earlier question, 
[Regions on a sphere that avoid a fixed point set](https://mathoverflow.net/q/208230/6094).
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<img src="https://i.sstatic.net/bLNKd.jpg" width="180" />
<br />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<sup>
A $5$-point set, its convex hull (blue) and smallest enclosing circle (red).
</sup>
<hr />
The OP's version is an $8$-point set. 
An upperbound was proved:
"for an $n$-point set $P$,
the area of an avoiding set cannot be larger
than $\frac{n-1}{n} A$, where $A$ is the area of the sphere."
For $n=8$, this gives an upper bound of $\frac{7}{8}=0.875$ which is smaller
than the OP's $0.9$.

To address the OP's specific questions:

> I want to know where this question originates from. 

My version was original, but ...

> Is this question part of some general type of questions that are encountered in a more general setting?

Yes, *avoiding sets*. For example, there is quite a bit of literature on *mutually avoiding sets*, going back to Erdős. There is also considerable literature on sets avoiding integral (or rational) distances. See, e.g.,
the question [Integer-distance sets](https://mathoverflow.net/q/136925/6094). Or:
Kurz, Sascha, and Valery Mishkin. "[Open sets avoiding integral distances](https://arxiv.org/abs/1204.0403)." *Discrete & Computational Geometry* 50.1 (2013): 99-123.