For years I've been using the following definition of the general position of a set of points in $\mathbb{R}^d:$
"The set of points in $\mathbb R^d$ is in general position iff every $(d+1)$ points are not in any possible hyperplane, i.e. no hyperplane contains more than $d$ points."
However, recently I have been reading Computational topology by Edelsbrunner and Harer, and their definition is the following:
"We say that the finite set $S \subset \mathbb{R}^d$ is in general position if no $d+2$ of the points lie on a common $(d-1)$-sphere."
These definitions are clearly different. To see it, take $d=2$ and consider the set of $4$ points forming a square. They satisfy the first definition, but not the second. Also, take $3$ colinear points. They satisfy the the second definition, but not the first.
So my question is: which definition is more common? Are they somehow related? Is it possible that one of these definitions is known under a different name?
Thank you for your time.
"General position" is a very broad term used in many different geometric and topological settings. Both the "planar" definition you're used to and the "spherical" definition you encountered more recently are special cases. Although it's probably too much to try to explain the very broadest generalizations of the term "general position", this (not very well written) wikipedia page on general position explains a general context into which both of your cases fit, under the heading "More generally".