Let $(Q,\pmb{g})$ be an $n$-dim Riemannian manifold with Levi-Civita connection/covariant derivative $\nabla$. I believe I understand the vertical and horizontal lifts a vector field on $Q$ to a vector field on $TQ$. However, I am confused about how the vertical and horizontal lifts are defined for 1-forms on $Q$.
For all of the following, let $(q^1,\dots,q^n):Q\to \mathbb{R}^n$ be local coordinates on the base with frame fields denoted $\pmb{e}_{q^i}\in\mathfrak{X}(Q)$ and $\pmb{\epsilon}^{q^i}\in\Omega(Q)$ (I'm saving the usual coordinate basis notation for $TQ$). Let $\Gamma^i_{jk}$ be the Christoffel symbols for this $q^i$-basis. Let $(q^1,\dots,q^n,v^1,\dots,v^n):TQ\to \mathbb{R}^{2n}$ be the corresponding local `natural' tangent bundle coordinates (i.e., tangent-lifted coordinates) with frame fields denoted $\pmb{\partial}_{q^i},\pmb{\partial}_{v^i}\in\mathfrak{X}(TQ)$ and $\pmb{d}q^i,\pmb{d}v^i\in\Omega(TQ)$.
First, a sanity check for the case of vector fields. Correct me if the following is wrong. The vertical and horizontal lifts of some $\pmb{w}=w^i\pmb{e}_{q^i}\in\mathfrak{X}(Q)$ are given in the $(q^i,v^i)$ coordinate basis by:
$$ \pmb{w}^V = w^i\pmb{\partial}_{v^i} \in\mathfrak{X}_V(TQ) \qquad,\qquad \pmb{w}^H = w^i \pmb{\partial}_{q^i} - \Gamma^i_{jk}v^jw^k \pmb{\partial}_{v^i} = w^i \pmb{D}_{q^i} \in\mathfrak{X}_H(TQ) $$
where $\pmb{D}_{q^i}=(\pmb{e}_{q^i})^H = \pmb{\partial}_{q^i} - v^j\Gamma^k_{ji}\pmb{\partial}_{v^k} $ are the local "horizontal basis vectors" such that $\pmb{D}_{q^i},\pmb{\partial}_{v^i}\in \mathfrak{X}(TQ)$ are "adapted to" the decomposition $TTQ= HTQ\oplus VTQ$.
My question: for some $\pmb{\alpha}=\alpha_i\pmb{\epsilon}^{q^i}\in\Omega(Q)$, what is the vertical and horizontal lifts? I am inclined to say they are as follows:
$$ \pmb{\alpha}^V = \Gamma^k_{ji}v^j\alpha_k \pmb{d}q^i + \alpha_i \pmb{d}v^i = \alpha_i\pmb{\Delta}^{v^i} \in \Omega_V(TQ) \qquad\qquad \pmb{\alpha}^H = \pi^*\pmb{\alpha} = \alpha_i \pmb{d}q^i \in \Omega_H(TQ) $$
where $\pi:TQ\to Q$ is the projection and $\pmb{\Delta}^{v^i} = (\pmb{\epsilon}^{q^i})^V = \pmb{d}v^i + v^j\Gamma^i_{jk}\pmb{d}q^k$ are `vertical basis 1-forms' satifying $\pmb{\Delta}^{v^i}\cdot \pmb{D}_{q^j} = 0$. I have seen some sources define the vertical and horizontal lifts of 1-forms the way I have them above. I have seen other sources define them in exactly the opposite way: that is, my above $\pmb{\alpha}^V$ is called the horizontal lift, and my above $\pmb{\alpha}^H$ is called the vertical lift. What is the correct terminology? Do we have a decomposition $T^*TQ= H^*TQ\oplus V^*TQ$ (I thnk that's what the above would imply?) or do we have $T^*TQ= V^*TQ\oplus H^*TQ$?
bonus question: If what I have done above is correct (or if it is wrong but correct if I swap vertical $\leftrightarrow$ horizontal for the case of 1-forms), how are the vertical/horizontal lifts of vector fields and 1-forms from $Q$ to $T^*Q$ defined?
Note: I'm not a mathemetician but, on my quest to teach myself geometric mechanics, I have stumbled into deeper mathematical waters than I intended. I'm aware that everything I did above can be formulated in a more rigorous, coordinate-agnostic, way. And I'm aware that this topic can be generalized well beyond the tangent bundle of a Riemannian manifold. But, right now, that's the case I am interested in, as well as the cotangent bundle of a Riemannian manifold as per my last question.