Here's one approach to formalizing "moved around". Let $G=(V,E)$ be a graph. Let $p:V\rightarrow S$ be a placement of $G$, that is, a map from the vertex set of $G$ to $S$. Let us call the data $(G,p)$ a framework on $S$. Let us define a motion of $(G,p)$ to be a continuous family of placements $f:V\times[0,1]\rightarrow S$ such that:

1. $f(v,0) = p(v)$ for all $v\in V$, that is, the motion begins at $p$.
2. $d_S(f(u,t),f(v,t))=d_S(p(u),p(v))$ for all $uv\in E$ and all $t\in[0,1]$, where $d_S(\cdot,\cdot)$ is the distance function on $S$. This condition just states that the motion preserves the lengths of all edges of $G$.
3. $\alpha_t(u,v,w)=\alpha_0(u,v,w)$ for all triples of vertices $uvw$ such that $uv\in E$ and $vw\in E$, where $\alpha_t(u,v,w)$ is the angle between the geodesic segments $uv$ and $vw$ at $v$. This condition ensures that the motion preserves all angles between adjacent pairs of edges of $G$.

[Such frameworks are related to the point-line frameworks of Jackson and Owen, and also work of Tay, Whiteley, Jackson and Jordán and others on 2D molecular graphs and frameworks (see e.g. this paper of Jackson and Jordán.]

Your question is essentially: Let $(G,p)$ be the molecular framework constructed from a geodesic triangle $T$ on $S$. Suppose there exists a motion from $(G,p)$ to any other congruent geodesic triangle (i.e. one with the same lengths, angles and orientation as $T$). Does this imply that $S$ has constant curvature?

I suspect the answer is no, for a possibly silly reason. It's not obvious to me that generic embedded surfaces need to have any pairs of distinct congruent geodesic triangles at all; the condition would be vacuous on such surfaces, which also don't have constant curvature. If we add an additional condition on $S$ that such pairs exist, then the answer seems like it might be yes, but I'm not up for proving it at this moment...

Here's one approach to formalizing "moved around". Let $G=(V,E)$ be a graph. Let $p:V\rightarrow S$ be a placement of $G$, that is, a map from the vertex set of $G$ to $S$. Let us call the data $(G,p)$ a framework on $S$. Let us define a motion of $(G,p)$ to be a continuous family of placements $f:V\times[0,1]\rightarrow S$ such that:

1. $f(v,0) = p(v)$ for all $v\in V$, that is, the motion begins at $p$.
2. $d_S(f(u,t),f(v,t))=d_S(p(u),p(v))$ for all $uv\in E$ and all $t\in[0,1]$, where $d_S(\cdot,\cdot)$ is the distance function on $S$. This condition just states that the motion preserves the lengths of all edges of $G$.
3. $\alpha_t(u,v,w)=\alpha_0(u,v,w)$ for all triples of vertices $uvw$ such that $uv\in E$ and $vw\in E$, where $\alpha_t(u,v,w)$ is the angle between the geodesic segments $uv$ and $vw$ at $v$. This condition ensures that the motion preserves all angles between adjacent pairs of edges of $G$.

[Such frameworks are related to the point-line frameworks of Jackson and Owen, and also work of Tay, Whiteley, Jackson and Jordán and others on 2D molecular graphs and frameworks (see e.g. this paper of Jackson and Jordán.]

Your question is essentially: Let $(G,p)$ be the framework constructed from a geodesic triangle $T$ on $S$. Suppose there exists a motion from $(G,p)$ to any other congruent geodesic triangle (i.e. one with the same lengths, angles and orientation as $T$). Does this imply that $S$ has constant curvature?

I suspect the answer is no, for a possibly silly reason. It's not obvious to me that generic embedded surfaces need to have any pairs of distinct congruent geodesic triangles at all; the condition would be vacuous on such surfaces, which also don't have constant curvature. So let us add an additional condition on $S$ that such pairs exist.

edit:

Thanks to the answer of Zurab Silagadze, I can see that a positive answer to a related question was claimed by Riemann in §II.4 of his famous paper "Ueber die Hypothesen welche der Geometrie zu Grunde liegen", (see also this English translation by Clifford). Here is an edited version of Clifford's translation of the passage in question:

The common character of manifolds with constant curvature may also be expressed thus, that figures may be moved in them without stretching. For clearly figures could not be arbitrarily shifted and turned round in them if the curvature at each point were not the same in all directions. On the other hand, however, the measure-relations of the manifold are entirely determined by the curvature; they are therefore exactly the same in all directions at one point as at another, and consequently the same constructions can be made from it: whence it follows that in manifolds with constant curvature figures may be given any arbitrary position.

It is not clear to me what "figures" are being considered here, and I admit to not understanding what exactly is proved here, if anything. According to §2.2 of Hans Freudenthal's "Lie groups in the foundations of geometry", the first proof was given by Rudolf Lipschitz in the 1870 paper Fortgesetzte Untersuchungen in Betreff der ganzen homogenen Functionen von n Differentialen. Unfortunately, my German is not up to the task of finding the precise statement in this paper. Any takers?

Here's one approach to formalizing "moved around". Let $G=(V,E)$ be a graph. Let $p:V\rightarrow S$ be a placement of $G$, that is, a map from the vertex set of $G$ to $S$. Let us call the data $(G,p)$ a framework on $S$. Let us define a motion of $(G,p)$ to be a continuous family of placements $f:V\times[0,1]\rightarrow S$ such that:

1. $f(v,t) = p(v)$ for all $v\in V$, that is, the motion begins at $p$.
2. $d_S(f(u,t),f(v,t))=d_S(p(u),p(v))$ for all $uv\in E$ and all $t\in[0,1]$, where $d_S(\cdot,\cdot)$ is the distance function on $S$. This condition just states that the motion preserves the lengths of all edges of $G$.
3. $\alpha_t(u,v,w)=\alpha_0(u,v,w)$ for all triples of vertices $uvw$ such that $uv\in E$ and $vw\in E$, where $\alpha_t(u,v,w)$ is the angle between the geodesic segments $uv$ and $vw$ at $v$. This condition ensures that the motion preserves all angles between adjacent pairs of edges of $G$.

[Such frameworks are related to the point-line frameworks of Jackson and Owen, and also work of Tay, Whiteley, Jackson and Jordán and others on 2D molecular graphs and frameworks (see e.g. this paper of Jackson and Jordán.]

Your question is essentially: Let $(G,p)$ be the molecular framework constructed from a geodesic triangle $T$ on $S$. Suppose there exists a motion from $(G,p)$ to any other congruent geodesic triangle (i.e. one with the same lengths, angles and orientation as $T$). Does this imply that $S$ has constant curvature?

I suspect the answer is no, for a possibly silly reason. It's not obvious to me that generic embedded surfaces need to have any pairs of distinct congruent geodesic triangles at all; the condition would be vacuous on such surfaces, which also don't have constant curvature. If we add an additional condition on $S$ that such pairs exist, then the answer seems like it might be yes, but I'm not up for proving it at this moment...

Here's one approach to formalizing "moved around". Let $G=(V,E)$ be a graph. Let $p:V\rightarrow S$ be a placement of $G$, that is, a map from the vertex set of $G$ to $S$. Let us call the data $(G,p)$ a framework on $S$. Let us define a motion of $(G,p)$ to be a continuous family of placements $f:V\times[0,1]\rightarrow S$ such that:

1. $f(v,0) = p(v)$ for all $v\in V$, that is, the motion begins at $p$.
2. $d_S(f(u,t),f(v,t))=d_S(p(u),p(v))$ for all $uv\in E$ and all $t\in[0,1]$, where $d_S(\cdot,\cdot)$ is the distance function on $S$. This condition just states that the motion preserves the lengths of all edges of $G$.
3. $\alpha_t(u,v,w)=\alpha_0(u,v,w)$ for all triples of vertices $uvw$ such that $uv\in E$ and $vw\in E$, where $\alpha_t(u,v,w)$ is the angle between the geodesic segments $uv$ and $vw$ at $v$. This condition ensures that the motion preserves all angles between adjacent pairs of edges of $G$.

[Such frameworks are related to the point-line frameworks of Jackson and Owen, and also work of Tay, Whiteley, Jackson and Jordán and others on 2D molecular graphs and frameworks (see e.g. this paper of Jackson and Jordán.]

Your question is essentially: Let $(G,p)$ be the molecular framework constructed from a geodesic triangle $T$ on $S$. Suppose there exists a motion from $(G,p)$ to any other congruent geodesic triangle (i.e. one with the same lengths, angles and orientation as $T$). Does this imply that $S$ has constant curvature?

I suspect the answer is no, for a possibly silly reason. It's not obvious to me that generic embedded surfaces need to have any pairs of distinct congruent geodesic triangles at all; the condition would be vacuous on such surfaces, which also don't have constant curvature. If we add an additional condition on $S$ that such pairs exist, then the answer seems like it might be yes, but I'm not up for proving it at this moment...