I came across this thread today when I was reading about almost quaternionic structures. I was wondering if there exists an argument similar to the answer to the thread above that can show that the $GL(n,\mathbb{H})\cdot \mathbb{H}^\times$-structure definitions of an almost quaternionic structure and the definition by a pair of almost complex structures $J, K : TM\to TM$ such that $J\circ K+K\circ J=0$ (in nLabs or Sommese).
I failed to find some similar condition that a subset of frames must satisfy which leads to a principal $GL(n,\mathbb{H})\cdot \mathbb{H}^\times$-subbundle of $\mathcal F(M)$. In all the text I found, this part of equivalence is either not mentioned as some authors (like Sternberg) focus on one of the definitions, or is considered as "obvious or easy" to the readers.
AS far as I understand, the second "definition" that you give is not equivalent to the definition as a $G$-structure. (This is visible from the text in nLab, which I don't find very clear however.) The standard definition in direct terms is via a rank 3 subbundle $Q\subset L(TM,TM)$ which has the property that, locally around each point, it can be spanned by $J$, $K$, and $J\circ K$ for two anti-commuting almost complex structures $J$ and $K$. However, $J$ and $K$ themselves are not part of the data (and there is an $SO(3)$-freedom in their choice in each point). Indeed if you require global almost complex structures $J$ and $K$, then you get to a hypercomplex structure which corresponds to the structure group $GL(n,\mathbb H)$.
The source of the difficulty is that quaternionic scalar multiplications are not quaternionically linear maps, since the quaternions are non-commutative, and they do move the standard quaternions $i$, $j$, $k$.
For the correct definition via $Q$, there is a similar description as in the question you link to. You form the linear frame bundle of $M$ using $\mathbb H^n$ as $\mathbb R^{4n}$. Then you can either define the reduction as consiting of all linear isomorphisms $\phi:\mathbb H^n\to T_xM$ such that for each $A\in Q_x$ the map $\phi^{-1}\circ A\circ\phi\in L(\mathbb H^n,\mathbb H^n)$ is given by scalar multiplication by some imaginary quaternion. Alternatively, you take distinguished elements $J(x), K(x)\in Q_x$, use them to make $T_xM$ into a quaternionic vector space and then consider the isomorphisms $\phi$ that can be written as a quaternionic scalar multiplication followed by a quaternionically linear map.