Understanding of Spin(n) and SO(n)

Question

I want to make sure I understand the relation between spin and rotation (mainly between SU(2) and SO(3), but also in general). (I am a physics major, so I apologize if my statements are not very rigorous, but I want to make sure I understand the basic underlying concepts.) So SU(2) is the double cover of SO(3). Also Spin(3) is the double cover of SO(3). So, SU(2) and Spin(3) are isomorphic. Now I am a bit confused about the objects that these groups act on. If I think of SU(2), they act (in the fundamental representation) on 2 dimensional objects, which are called spinors. Now, if I understand it right, when labeling the representations of SU(2) by j (the value of the angular momentum), if j is half integer it is a spinorial representation, while if j is integer it is a vectorial representation. So for j=1, the object acted upon are 3 dimensional vectors, not spinors? But as we are in SU(2), which are complex matrices, the vectors are complex vectors? So the difference between a complex vector and a tensor is given by the representation to which they belong to? So a 3 dim object which changes under a 3x3 representation of SU(2) is a complex vector, while a 2 (or 4, 6 etc) dimensional object changing under a 2x2 (4x4, 6x6 ..) representation of SU(2) is a spinor? Or is it anything deeper that this? Now if we go to higher spin, lets say that a k dimensional object changes under the k-dim representation of Spin(n) and another k-dim object under k-dim representation of Spin(m). Do we decide whether they are spinors or not based on whether that representation is spinorial or vectorial? I.e. a k-dim object on its own can't be called a complex vector or a spinor, unless we know how it transforms? Please let me know if what I said is wrong, and how should I think about all these? Thank you!