I've been researching spinors, and I'm a bit confused by some of the terminology. In some cases, spinors seem to be presented as elements of the Spin group, whereas in others they seem to be presented as "vector-like" objects that transform under the Spin group (the latter seeming to be more common in physics settings).
Which description is more accurate? Is this a case of "overloaded" terminology referring to two different (albeit related) objects? Or are the two equivalent, and the distinction is irrelevant?
Lastly, if spinors are in fact elements of the Spin group—and thus also of the relevant Special Unitary group—how does one reconcile this with the way spinors are typically depicted in physics? In 3 dimensions, for example, how would one reconcile the concept of spinors as elements of SU(2) with their usual depiction as two-component complex column "vectors"?
[Apologies if this question is too "physics-y"—I'm mainly interested in the pure math perspective here, but I'll repost to the Physics SE if it's not appropriate for this forum]