There are a number of mathematical imprecisions in your question and your answer. Some advice: you will be less confused if you take more care to avoid sloppy language.
First, the term spinor either refers to the fundamental representation of $SU(2)$ or one of the several spinor representations of the Lorentz group. This is an abuse of language, but not a bad one.
A particularly fussy point: What you've described in your first paragraph is a spinor field, i.e., a function on Minkowski space which takes values in the vector space of spinors.
Now to your main question, with maximal pedantry: Let $L$ denote the connected component of the identity of the Lorentz group $SO(3,1)$, aka the proper orthochronous subgroup. Projective representations of $L$ are representations of its universal cover, the spin group $Spin(3,1)$. This group has two different irreducible representations on complex vector spaces of dimension 2, conventionally known as the left- and right- handed Weyl representations. This is best understood as a consequence of some general representation theory machinery.
The finite-dimensional irreps of $Spin(3,1)$ on complex vector spaces are in one-to-one correspondence with the f.d. complex irreps of the complexification $\mathfrak{l}_{\mathbb{C}} = \mathfrak{spin}(3,1) \otimes \mathbb{C}$ of the Lie algebra $\mathfrak{spin}(3,1)$ of $Spin(3,1)$. This Lie algebra $\mathfrak{l}_{\mathbb{C}}$ is isomorphic to the complexification $\mathfrak{k} \otimes \mathbb{C}$ of the Lie algebra $\mathfrak{k} = \mathfrak{su}(2) \oplus \mathfrak{su}(2)$. Here $\mathfrak{su}(2)$ is the Lie algebra of the real group $SU(2)$; it's a real vector space with a bracket.
I'm being a bit fussy about the fact that $\mathfrak{su}(2)$ is a real vector space, because I want to make the following point: If someone gives you generators $J_i$ ($i=1,2,3$) for a representation of $\mathfrak{su}(2)$, you can construct a representation of the compact group $SU(2)$ by taking real linear combinations and exponentiating. But if they give you two sets of generators $A_i$ and $B_i$, then you by taking certain linear combinations with complex coefficients and exponentiating, you get a representation of $Spin(3,1)$, aka, a projective representation of $L$. If memory serves, the 6 generators are $A_i + B_i$ (rotations) and $-i(A_i - B_i)$ (boosts). See Weinberg Volume I, Ch 5.6 for details.
The upshot of all this is that complex projective irreps of $L$ are labelled by pairs of half-integers $(a,b) \in \frac{1}{2}\mathbb{Z} \times \frac{1}{2}\mathbb{Z}$. The compex dimension of the representation labelled by $a$,$b$ is $(2a + 1)(2b+1)$.
The left-handed Weyl-representation is $(1/2,0)$. The right-handed Weyl representation is $(0,1/2)$. The Dirac representation is $(1/2,0)\oplus(0,1/2)$. The defining vector representation of $L$ is $(1/2,1/2)$.
The Dirac representation is on a complex vector space, but it has a subrepresentation which is real, the Majorana representation. The Majorana representation is a real irrep, but in 4d it's not a subrepresentation of either of the Weyl representations.
This whole story generalizes beautifully to higher and lower dimensions. See Appendix B of Vol 2 of Polchinski.
Figuring out how to extend these representations to full Lorentz group (by adding parity and time reversal) is left as an exercise for the reader. One caution however: parity reversal will interchange the Weyl representations.
Sorry for the long rant, but it raises my hackles when people use notation that implies that some vector spaces are spheres. (If it's any consolation, I know mathematicians who get very excited about the difference between a representation $\rho : G \to Aut(V)$ and the "module" $V$ on which the group acts.)
Perhaps the shortest answer (from my elementary understanding of representation theory) would be that the differences between the types of spinors you asked for lie primarily in terms of the representations of the rotation group under which they transform: If the spinors are symbolised by ψ, then the transformation rule:
Ψ’ = M Ψ, where M is one of the matrix reps of the rotation group.
will have:
M = SL(2,C) for 2-component Weyl (and other relativistic, e.g. Lorentz) spinors, which obey the Weyl equation (the massless form of the Dirac equation)
M = SU(2) for the non-relativistic 2-component (Pauli) spinors, which obey the Schrodinger-Pauli equation – the non-relativistic but massive limit of the Dirac equation (I suspect their components, for normalised wave functions are each restricted to the unit circle in the complex plane.)
M = something much more general than either of the above for Cartan spinors in their most general form (possibly via SO(p,q) for general p, q?) as Cartan claimed his spinors are the most general mathematical form of spinors, and they deal with rotations in spaces of any number of dimensions. They should therefore range beyond 2-component objects (as do even Weyl spinors for e.g. 6-D space).
Although you profess disinterest in Dirac and Majorana spinors you might also like to refer to a comparable type (but much more expert) answer comparing Weyl with Dirac and Majorana spinors.
Meanwhile, as others can probably provide a better answer, I regard this as an opportunity to learn more by inviting corrections; otherwise, I might post a more detailed answer, with references, later.
UPDATE: Abstract of the ref given as this paper in Physics StackExch 381625 states:
"...The physical observables in Schrödinger–Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit." (behind a paywall).
Best Answer
The spin group has a multiple irreducible representations of dimension 4. Two of them are the left- and right-handed spin-3/2 representations. The other one is called the vector representation. "Ordinary" 4-vectors belong to (copies of) the latter representation, which is not considered one of the "spinor representations". Elements of these representations do all "rotate differently" from each other under elements of the spin group. In general, a representation of the spin group is labelled $(p,q)$ for $p,q$ nonnegative half-integers, with $n=p+q$ being the spin and the dimensionality being $(2p+1)(2q+1).$
In math, "vector" just means "element of a vector space". Spinors are by definition always "vectors" in the math sense. But in physics, "vector" usually means more than it does in math. Intuitively, a physical vector is "a magnitude with a direction" in space(time). An element of $\mathbb R^{100}$ is not going to be a physical vector. Even an element of $\mathbb R^4$ should only be considered a physical vector if you've fixed a coordinate system etc. to interpret it as a magnitude and a direction in spacetime. In fact, one structure you'd like to see on any space of physical vectors is a vector representation of the spin group on it, which tells you how to rotate the elements. (Again, the vector representation is the $\left(\frac12,\frac12\right)$ one, not the $\left(\frac32,0\right)$ or $\left(0,\frac32\right)$ spinor ones.) Hopefully this clarifies why in physics we distinguish "vectors" from "spinors".
Note that treating a spinor $v$ as a pair $(v,\rho)$ where $\rho$ is the representation is much like treating a (math) vector $v$ as a pair $(v, V)$ where $V$ is a vector space. It's somewhat wrong-headed. The word "vector" in math means "member of a vector space" and says nothing about what the object actually is. If you want to talk about vectors, you first fix your vector space and then you talk about its elements. The same is true for spinors. If you want to talk about spinors, you pick some spinor representation, perhaps postulated by some physical theory, and then you start talking about its elements. You don't consider the representation to be part of the spinor; it's the other way around.
The Dirac equation is a differential equation for a field $\psi:M^{3,1}\to V,$ where $V$ is the space of "Dirac spinors". Specifically, we write $V=\left(\frac12,0\right)\oplus\left(0,\frac12\right)$ to denote that $V$ is a four-dimensional space carrying a reducible representation of the spin group, where two of the dimensions transform under the (irreducible) $\left(\frac12,0\right)$ representation (left-handed spin-1/2) and the other two independently transform under the $\left(0,\frac12\right)$ irrep (right-handed spin-1/2), such that $V$ is a direct sum of two smaller spaces with the specified representations.