Why is a positive cone defined the way it is?

Question

Let $K$ be a field. Consider $P \subseteq K$ satisfying the following properties:

1. $x \in P$ and $y \in P$ imply $x+y \in P$ and $xy \in P$

2. $x \in K$ imply $x^2 \in P$

3. ${-1} \notin P$

We call such $P$ a prepositive cone. We further call $P$ a positive cone if $K = P \cup {-P}$.

I have a few questions regarding the notion of a positive cone (I have not found references for this concept elsewhere other than Wikipedia).

1. Why the name positive "cone"? I understand why positive is there, but not the "cone" part.
2. If we keep Property $1$, delete Property $2$ and Property $3$, and add Property $2'$, which says that $x \in P$, or $x = 0$, or $x \notin P$ (only one case holds for any $x \in K$), and define a positive cone this way, what do we lose? Why not define positive cones this way instead?