Is it absolutely rigorously true that each topological manifold of ${\rm dim} \lt 4$ admits exactly only one differential structure?

Question

I'm struggling with statements such as the following in Wikipedia:

"[In] dimensions smaller than 4, there is only one differential structure for each topological manifold.
That was proved by Tibor Radó for dimension 1 and 2, and by Edwin E. Moise in dimension 3."

This certainly seems a perfectly clear-cut, incontrovertible statement; and I certainly have neither intention nor expertise to argue against the mentioned proofs.

But still ... I wonder whether perhaps there might be certain exceptions or limitations which seem "so obvious, as to not even be worth mentioning" to anyone who'd competently make and agree with the above statement; perhaps considerations that "go without saying" by experts.

For example, there is certainly one caveat concerning distinction and counting of differential structures:
it's meant "only up to equivalence by diffeomorphism". (To put it simply ...)

Now, I do have a specific question (based on section Existence and uniqueness theorems of the same Wikipedia page) which is motivated by my suspicion of another ("unwritten") caveat to the statement quoted above:

For simplicity and concreteness, consider a $2$-dimensional topological manifold $(\mathcal S, \mathsf S)$ given in terms of $2$-dimensional topological space $\mathcal S$ and a (presumably unique) maximal topologically compatible atlas $\mathsf S$.

Included as a subset shall be one specific $2$-dimensional (maximal) $C^1$-atlas $\mathsf P^{(1)} \subset \mathsf S$ (which in turn, if I understand correctly, has as subsets some (non-zero) number of distinct but mutually diffeomorphic $2$-dimensional (maximal) $C^{\infty}$-atlases, $\mathsf A_j^{(\infty)} \subset \mathsf P^{(1)}$. Jointly they count as exactly one differential structure).

My question:

Does atlas $\mathsf S$ then also include (at least) one other specific $2$-dimensional (maximal) $C^1$-atlas $\mathsf Q^{(1)} \subset \mathsf S$, which is $\mathsf Q^{(1)} \ne \mathsf P^{(1)}$, such that overlapping charts of $\mathsf Q^{(1)}$ and $\mathsf P^{(1)}$ are not at each point of the respective overlap region differentiable wrt. each other (and where there exists at least one $2$-dimensional (maximal) $C^{\infty}$-atlas $\mathsf B^{(\infty)} \subset \mathsf Q^{(1)}$, which is $\mathsf B^{(\infty)} \not\subset \mathsf P^{(1)}$) ?

If so, are these two distinct (maximal) $C^1$-atlases $\mathsf P^{(1)}$ and $\mathsf Q^{(1)}$ indicative of corresponding two distinct differential structures being attached to the same one topological manifold $(\mathcal S, \mathsf S)$ ?