General elliptic K3 surfaces. Consider K3 surfaces of Picard rank two with Neron-Severi lattice isomorphic to $$\left[\begin{array}{cc} 2d & t \\ t & 0 \end{array}\right]$$ for some positive integers $d, t$. Such K3 surfaces are necessarily elliptic (because they have class of square zero), and conversely any projective elliptic K3 surface of Picard rank two has such a Neron-Severi lattice. Here $t$ is the minimal degree of a multisection (which is uniquely defined) and $2d$ is the degree of polarization (which is only well-defined mod. $t$). Elliptic surfaces with a section correspond to the case $t = 1$: for any $d$ such lattice is isomorphic to the hyperbolic plane $U$.

The moduli space. By the Torelli theorem for K3 surfaces, for fixed $d, t > 0$ such K3 surfaces exist and they form an $18$-dimensional moduli space $\mathcal{M}_{d,t}$. (These elliptic surfaces have either one or two elliptic fibrations depending on $d, t$.)

Singular fibers. Assume that singular fibers of the elliptic fibrations have a single node (type $I_1$) so that by the Euler characteristic count there are $e(X) = 24$ singular fibers.

Question: do these $24$ points determine the general elliptic K3 from $\mathcal{M}_{d,t}$ up to finitely many choices? If not, what can be said about the dimension of the moduli space of those elliptic K3 surfaces with a prescribed set of $24$ singular fibers?

Possibly if one deals with the $\mathcal{M}_{0,1}$ (elliptic surfaces with a section) the general case will follow by considering the Jacobian fibration (as suggested by user25309 in the comments).

Any suggestions or references welcome.