Comments to the question (v3):
I) The notions of vectors, tensors, scalars, etc, depend on contexts in physics, cf. e.g. this and this Phys.SE posts and links therein.
II) In OP's context, these notions refer to representations $\rho$ of the Lie group $SO(3)$ [and the corresponding Lie algebra $so(3)$] of 3D rotations, cf. e.g. Ref. 1. Let $\mathrm{i}L_k$, $k=1,2,3$, denote the $3$ antisymmetric real $3\times 3$ matrix generators of $so(3)$:
$$\mathrm{i}(L_k)_{ij}~:=~\epsilon_{ijk}.\tag{A}$$
In this context, a vector$^1$ $V_k$ is an object that transforms in the 3-dimensional vector (or triplet) representation $\bf 3$ of $SO(3)$. Concretely this means that $$(\rho(\mathrm{i}L_k)V)_i~=~\epsilon_{ijk}V_j,\tag{B}$$
up to sign conventions. Similarly, a scalar $S$ is an object that transforms in the 1-dimensional trivial (or singlet) representation $\bf 1$ of $SO(3)$. Concretely this means that $$(\rho(\mathrm{i}L_k)S)_i~=~0.\tag{C}$$
Example: OP's vector field (2) can be viewed as an element in the vector representation of $SO(3)$. Not surprisingly, the group action of a rotation $R\in SO(3)$ on the vector field leads to rigid rotations of the vector field (2). However, the vector field (2) is not a dynamical active vector, see next section III.
III) In the Hamiltonian formalism, for objects constructed out of e.g. $x_i$, $p_j$, $\delta_{ij}$, $\epsilon_{ijk}$, etc., and contractions thereof, there is also another notion. Let us for the purpose of this answer call it the notion of dynamical active vectors, tensors, scalars, etc. E.g. a dynamical active vector $V_k$ satisfies by definition
$$ \{L_i,V_j\}_{PB}~=~\epsilon_{ijk}V_k\tag{D}$$
where
$$L_i~:=~\epsilon_{ijk}x_jp_k\tag{E}$$
is the orbital angular momentum, and
$$\{x_i,p_j\}_{PB}~=~\delta_{ij}~=~-\{p_i,x_j\}_{PB},\quad \{x_i,x_j\}_{PB}~=~0~=~\{p_i,p_j\}_{PB},\tag{F}$$
is the canonical Poisson brackets (PB). Similarly, a dynamical active scalar $S$ satisfies
$$ \{L_i,S\}_{PB}~=~0.\tag{G}$$
Example 1: Consider the inner/dot product $S=a_kx_k$, where $a_k$ is a constant (dynamically passive) vector. Then $S$ is a scalar (C), but $S$ is not a dynamical active scalar (G).
Example 2: The inner/dot products $S=x_kx_k$, $S=x_kp_k$, $S=p_kp_k$, and $S=a_ka_k$ are all dynamical active scalars according to definition (G).
Example 3: A linear combination
$$V_k~=~\sum_a S_{(a)} V_{(a)k},\tag{H}$$
of dynamical active vectors $V_{(a)k}$ with dynamical active scalar coefficients $S_{(a)}$ is again a dynamical active vector (D).
Example 4: The three objects $x_i$, $p_j$, and $L_k$ are all dynamical active vectors according to definition (D). In fact, these three objects constitute a basis for dynamical active vectors, in the sense of eq. (H), cf. Ref. 2 and OP's eq. (3).
References:
G. 't Hooft, Introduction to Lie Groups in Physics, lecture notes; chapter 3 + chapter 9. The pdf file is available here.
L.D. Landau & E.M. Lifshitz, Mechanics, vol. 1, 1976; $\S$42. Problem 3 & 4. Note that L&L have the opposite convention for the Poisson bracket (PB) as compared to eq. (F).
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$^1$ We assume here that the metric $g_{ij}=\delta_{ij}$, and hence will not distinguish between covariant and contravariant indices, cf. e.g. this Phys.SE post. Also we assume Einstein summation convention, i.e. summation over repeated indices.
