Weak amalgamation property for type space functors Definition 1.1 in the paper Omitting Types and the Baire Category Theorem gives the notion of a type space functor as a functor $S$ from $\textbf{FinSet}^{\text{op}}$ to the category of topological spaces with continuous open maps, taking each $n \in \textbf{FinSet}$ to a topological space $S_n$, and satisfying a "weak amalgamation property".
In detail, this weak amalgamation property reads as follows:

Let $i_k : k \to k+1$ be the inclusion and $d_m : m+1 \to m+2$ be given by $d(j) = j$ for $j <m$ and $d(m) = m+1$.  For each $m \in \textbf{FinSet}$, $p \in S_m$, $q \in (Si_m)^{-1}(\{p\})$, and non-empty open $U \subseteq (Si_m)^{-1}(\{p\})$, let $WAP_S(m,p,q,U)$ be the statement that there is $r \in S_{m+2}$ such that $(Si_{m+1})(r)=q$
and $(Sd_m)(r) \in U$.

The motivating example of this definition is the type space functor of a first-order theory $T$ (Definition 1.5), defined by sending each $n$ to the space of complete $n$ types of $T$ and each function $f : n \to m$ to the map $Sf : S_m(T) \to S_n(T)$ given by $$(Sf)(p) = \{\varphi(x_0, \dots, x_{n-1}) : \varphi(x_{f(0)}, \dots, x_{f(n-1)}) \in p\};$$ moreover, note that if $(a_0, \dots, a_{m-1})$ is a realization of a type $p \in S_m(T)$ in a model $\mathscr M \models T$, we have that $$(Sf)(p) = \text{tp}^{\mathscr M}(a_{f(0)}, \dots, a_{f(m-1)}).$$ Proposition 1.6 shows that this is indeed a type space functor as defined above.
Question. Has this weak amalgamation property defining type space functors anything to do with weak amalgamation as seen in the Model Theory literature?
As far as I'm aware (e.g. Definition 3.3 in I, Definition 4.1 in II, Definition 3.4.1 in III), we say that a class of structures $\mathcal K$  has the weak amalgamation property if for all $A \in \mathcal K$ there is an embedding $h: A \to B$ such that for all embeddings $f: B \to C_1$ and $g: B \to C_2$ there exist embeddings $f': C_1 \to D$ and $g': C_2 \to D$ such that $f' \circ f \circ h=g' \circ g \circ h$; the issue is that I'm not sure if this notion plays any role in the definition of type space functor which I might have overlooked or if it's just an unfortunate coincidence of terminology.

The only kind of amalgamation (but not strictly "weak" as above) that I could see in the case of type space functors for a first order theory $T$ is the following, extracted from the proof of Proposition 1.6; I'm skipping some details from the proof to ease clarity and I highlight in green the elements which I think take part in this amalgamation.
Fix $m \in \textbf{FinSet}$, $p \in S_m$, $q \in (Si_m)^{-1}(\{p\})$, and non-empty open $U \subseteq (Si_m)^{-1}(\{p\})$ and let $\color{green}{(a_0, \dots, a_m)}$ be a relization of $q$ in some model $\mathscr M \models T$. We can assume that $U = [\psi(x_0, \dots, x_m)]$ for some formula $\phi$; then $\varnothing \neq U \subseteq (Si_m)^{-1}(\{p\})$ implies $\exists y \psi(x_0, \dots, x_m, y) \in p$, and $q \in (Si_m)^{-1}(\{p\})$ implies that $\color{green}{(a_0, \dots, a_{m-1})}$ realizes $p$ in $\mathscr M$, so there is $a_{m+1}\in M$ such that $\mathscr M \models \psi\color{green}{(a_0, \dots, a_{m-1}, a_{m+1})}$. Letting now $r = \text{tp}^{\mathscr M}\color{green}{(a_0, \dots, a_{m-1}, a_m, a_{m+1})}$ gives that $(Si_{m+1})(r)=q$ and $(Sd_m)(r)\in U$, which verifies the $WAP_S(m,p,q,U)$ statement for the type space functor of the first order theory $T$.
From the definition of $i_m$, $i_{m+1}$ and $d_m$ one can check that:

*

*$(Sd_m)(r) = (Sd_m)(\text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_m, a_{m+1})) = \text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_{m+1})$.

*$(Si_{m+1})(r) =(Si_{m+1})(\text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_m, a_{m+1})) = \text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_m) = q$.

*$(Si_m)(\text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_{m+1}))= (Si_m)(\text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_{m}))= \text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}) = p$
Thus, for all $m \in \textbf{FinSet}$, $p \in S_m$, $q \in (Si_m)^{-1}(\{p\})$, and non-empty open $U \subseteq (Si_m)^{-1}(\{p\})$, if $(a_0, \dots, a_m)$ is a realization of $q$ in a model $\mathscr M \models T$, then there is $a_{m+1} \in M$ such that $$(Si_m)((Sd_m)(\text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_m, a_{m+1}))) = (Si_m)((Si_{m+1})(\text{tp}^{\mathscr M}(a_0, \dots, a_{m-1}, a_m, a_{m+1}))).$$
So that two $(m+1)$-types coinciding in the initial $m$-subtuple can be amalgamated into an $(m+2)$-type. This would explain why the authors refer to this property as "weak amalgamation" instead of "amalgamation" as Ben-Yaacov does in this paper, where he defines the (finite) amalgamation property for type space functors whenever (finite) types concide in any subtuple.
 A: As Alex's comment already mentions: the weak amalgamation for type space functors does not really have anything to do with the weak amalgamation property for classes of structures. It's just about being able to amalgamate types in a weak sense.
So let's make sense of that. Starting with Ben-Yaacov's definition of the amalgamation property for type space functors. Let me recall it here.

Definition. A type space functor $S$ has the amalgamation property if for any two finite sets $a$ and $b$ the natural map $S_{a \cup b} \to S_a \times_{S_{a \cap b}} S_b$ is surjective.

So what this is saying is that if we have a type $q_1 \in S_a$ and $q_2 \in S_b$, such that their restrictions to $a \cap b$ coincide, then there is a type $r \in S_{a \cup b}$ such that its restriction to $a$ is $q_1$ and its restriction to $b$ is $q_2$. So we have really amalgamated compatible types $q_1$ and $q_2$ to form a type $r$.
In the definition you quote we do not amalgamate types, we amalgamate a type and an open set (which, as you mention, we can think of as a formula). That is, the input is $q \in S_{m+1}$ and some open set $U \subseteq S_{m+1}$ that are compatible in the sense that $\{S_{i_m}(q)\} = S_{i_m}(U)$ (note that the $p$ is superfluous data). The definition then asks that we can find $r \in S_{m+2}$ such that $S_{i_{m+1}}(r) = q$ and $S_{d_m}(r) \in U$.
So if we take $a = m+1 = \{0, \ldots, m\}$, $b = \{0, \ldots, m-1, m+1\}$, $q_1 = q$ and we pick $q_2 \in U$ then we see that Ben-Yaacov's definition implies this weaker version. The converse would hold if $\{q_2\}$ is open. Of course, this is generally not the case, which is why this is truly a weaker version.
Edit: in an earlier version I wrote "$S_{i_m}(q) \in S_{i_m}(U)$" instead of "$\{S_{i_m}(q)\} = S_{i_m}(U)$" which is not quite equivalent to the definition. As Alex points out in the comments below, there is something odd about this definition, because it implies that $p = S_{i_m}(q)$ is isolated ($S_{i_m}$ is required to be an open map in the linked paper). This would mean that they only require amalgamation over isolated types (which is possible). My earlier version would then be a version of that definition that allows weak amalgamation over every type.
