A smooth manifold $M$ can be defined as a pair (topological manifold $X$, smooth, as in $C^{\infty}$, atlas $\mathscr M$), where a topological manifold is defined as a locally Euclidean Hausdorff 2nd countable space $X$ without regard to any continuous, as in $C^0$, atlas.
I want to know if I can define a smooth manifold $M$ instead as a pair (locally Euclidean Hausdorff 2nd countable space $X$, smooth atlas $\mathscr M$), and then a topological manifold $Y$ as a pair (locally Euclidean Hausdorff 2nd countable space $X$, $C^0$ atlas $\mathscr X$).
I mean, if I define a smooth manifold $M$ as a pair (topological manifold $Y$, smooth atlas $\mathscr M$), where $Y$ in turn is (locally Euclidean Hausdorff 2nd countable space $X$, continuous atlas $\mathscr X$) ... then does $M$ really does care about $\mathscr X$ ?
If so, then please give an example of smooth manifolds with the same smooth atlas (and topological space) but different continuous atlases and then they're indeed not identical or diffeomorphic or whatever? (I forgot the term, but I think it's like ... they're not considered smooth submanifolds of each other. Oh I think the term is 'distinct' or not 'identical' in the sense that the identity map is not the diffeomorphism or something. I think that's the 'equal underlying set' version of 'if the inclusion map is not a smooth immersion & topological embedding or something'.)
Like $M_1$ = ((locally Euclidean Hausdorff 2nd countable space $X$, continuous atlas $\mathscr X_1$), smooth atlas $\mathscr M$)
and
$M_2$ = ((locally Euclidean Hausdorff 2nd countable space $X$, continuous atlas $\mathscr X_2$), smooth atlas $\mathscr M$)
with $\mathscr X_1 \ne \mathscr X_2$ but then $M_1$ & $M_2$ are not diffeomorphic or whatever.
