defining smooth functions on smooth manifolds

If we're looking at the local version, the correct criterion for arbitrary charts is

For all charts $(V,\theta)$ with $p\in V$ there is an open neighbourhood $\tilde{V}\subset V$ of $p$ such that $f\circ \theta^{-1}$ is smooth on $\theta(\tilde{V})$.

Then it is easily seen that the existence of one chart $(U,\varphi)$ such that $f\circ \varphi^{-1}$ is smooth on $\varphi(\tilde{U})$ for some open neighbourhood $\tilde{U}$ of $p$ is sufficient to have the same for every chart around $p$, for if $(V,\theta)$ is a chart around $p$, then $\tilde{V} := \tilde{U} \cap V$ is an open neighbourhood of $p$, and on $\theta(\tilde{V})$

$$f\circ \theta^{-1} = (f\circ \varphi^{-1})\circ (\varphi\circ \theta^{-1})$$

is the composition of two smooth functions.

We cannot demand that $f\circ\theta$ is smooth on all of $\theta(V)$ then, since a chart around $p$ might contain points where $f$ is not smooth (possibly not even continuous).

The global variant follows from the local variant, since we can for every $x\in V$ choose a chart $(U_x,\varphi_x)$ around $x$ such that $f\circ \varphi_x^{-1}$ is smooth, from which it follows that $f\circ \theta^{-1}$ is smooth on a neighbourhood of $\theta(x)$, and since $x\in V$ was arbitrary, it then follows that $f\circ\theta^{-1}$ is smooth on all of $\theta(V)$.