In this answer a weak $2$-tree is a simplicial complex in which any two points can be separated by a forest. It is clear that a strong $2$-tree is also a weak $2$-tree. By passing to subdivisions, we may assume that any (specific) given subpolyhedron of a simplicial complex is a subcomplex, and we will do this implicitly when discussing trees in $2$-trees.

Define the face-edge graph $G$ of a $2$-dimensional simplicial complex $X$ to be the following graph: its vertices are the $2$- and $1$- dimensional faces of $X$ (so just triangles and edges). There is an edge in $G$ between a $2$-face $\sigma$ and a $1$-face $\tau$ if $\tau\subset\sigma$.

A free face in a simplicial complex $X$ is a face $\tau$ which is properly contained in precisely one face $\sigma$ (which must then be of dimension exactly $1+\dim\tau$). If $\sigma$ has a free face $\tau$ of codimension $1$, an elementary collapse can be performed which deletes both $\tau$ and $\sigma$: it is convenient to think of this as a strong deformation retract of $X$ onto the complement of $\sigma$. A complex which, by a sequence of elementary collapses, can be turned into a vertex is called collapsible. If some subdivision is collapsible the complex is called geometrically collapsible. It makes sense to speak of collapsing a complex onto a subcomplex, or geometrically collapsing a complex $X$ onto a subpolyhedron $Y$, and we will denote the situation that $X$ (geometrically) collapses onto $Y$ by $X \searrow Y$. (These notions are more or less standard in PL topology / combinatorial topology.) We may subdivide our complexes as much as convenient, although we won't really need to subdivide much. I won't keep track of when a certain sequence of collapses requires / does not require subdivisions.

Given a collapse $X\searrow Y$, we can realize each elementary collapse as a PL map in a more-or-less obvious way. Composing these we obtain a $\pi:X\rightarrow Y$ which I will call the ``collapse map'' below. This may not be unique -- there are probably several reasonable ways to define the map for each elementary collapse -- but it is a strong deformation retract with the property that the preimage of any point is a tree.

Claim: Let $X$ be a weak $2$-tree. If $\sigma\in X$ is a $2$-face, its component in $G$ is either infinite or contains a free edge of $X$.

Proof: Suppose the component of $\sigma$ in $G$ is finite and denote by $Y$ the union of all $1$- and $2$- faces in this component (including their vertices). Let $x,y$ be two points in the interior of $\sigma$. Let $F\subset X$ be a forest separating $x$ from $y$. Then $F^{\prime}=F\cap Y$ is a forest which separates $x$ from $y$ within $Y$. If a leaf of $F^{\prime}$ is not on a free edge, there is a path ``around'' this leaf, and it is redundant: The leaf and its unique edge can be deleted from $F^{\prime}$, and the resulting forest still separates $x$ from $y$ within $Y$. Deleting all redundant leaves iteratively in this way until none remain, we end up with a nontrivial forest, and each of its leaves is in a free edge of $Y$. ${\scriptstyle \blacksquare}$

Claim: A subcomplex of a weak $2$-tree is a weak $2$-tree.

Proof: Just intersect any forest which separates two given points with the subcomplex. It still intersects all paths between the points, and hence separates. ${\scriptstyle \blacksquare}$

Claim: A finite weak $2$-tree collapses onto a $1$-dimensional subcomplex.

Proof: Iteratively collapse all $2$-faces with a free edge. By the previous two claims, at each step we still have a finite weak $2$-tree, and there is still a free edge. ${\scriptstyle \blacksquare}$

Claim: A finite strong $2$-tree is collapsible.

Proof: Let $X$ be a finite strong $2$-tree. Collapse it onto a $1$-dimensional subcomplex $Y$ (a graph), and let $\pi:X\rightarrow Y$ be a corresponding PL retract satisfying that the preimage of any point is a tree. Assume there is a nontrivial loop, and let $x,y$ be two points on this loop. Their $\pi$-preimages are disjoint trees $T_{1},T_{2}$, which can be separated by a tree $T\subset X$. But then $\pi\left(T\right)$ is a connected set which separates $x$ from $y$, a contradiction. ${\scriptstyle \blacksquare}$

For lack of time to think and write in detail, I decided to include some ideas without proof. I would have preferred not to put something so "unripe" here, hopefully it is useful.

I think we can prove the following claim by induction on the number of faces (the induction step involves examining an elementary collapse performed in reverse).

Claim(?): A collapsible $2$-dimensional complex is a strong $2$-tree.

I will leave this for another time or for another person. I think it's easy if true (and probably true) but I did not check carefully, there could be some weird edge-case.

It is interesting is to ask what happens in the case of an infinite complex such as a triangulation of $\mathbb{R}^{2}$. This is probably necessary if we want to work with quasi-isometry... I suggest calling an infinite complex $X$ collapsible if there is an infinite chain $U_{1}\supseteq U_{2}\supseteq U_{3}\supseteq\ldots$ of open neighborhoods of the set of all ends of $X$ such that $X\setminus U_{i}$ is compact and collapsible for each $i$ and $\bigcap_{i}U_{i}=\emptyset$.

I think I have a proof that any (possibly infinite) strong $2$-tree is collapsible using this notion (perhaps I will update this answer with it on some other evening). I have not seen the definition before, perhaps it is new.

There is a decent chance these ideas extend to the higher dimensional notions.