In Example 16.9 of John Lee's Introduction to Smooth Manifolds 2nd edition, the following 2-form on $\mathbb{R}^3$ was considered: $$ w=x dy \wedge dz + y dz \wedge dx + z dx \wedge dy. $$ The example then considered the integration $$ \int_M \omega $$ on the manifold $M\subset{\mathbb{R}^3}$ which is the 2-sphere with radius 1.
My question is: wouldn't the integral only make sense if $\omega$ were a differential form on $M$ according to the definition? Here $\omega$ is defined on the ambient space $\mathbb{R}^3$.
I could only make sense of it in the following way. Take the first term $x dy\wedge dz$ as an example. Consider the coordinate charts $$\varphi_1: U=\{(y,z)\in \mathbb{R}^2, y^2+z^2<1 \} \mapsto M_1=\{(x,y,z)\in \mathbb{R}^3, x^2+y^2+z^2=1, x>0 \},\\ \varphi_1(y,z)=(\sqrt{1-y^2-z^2},y,z)$$ and $$\varphi_2: U=\{(y,z)\in \mathbb{R}^2, y^2+z^2<1 \} \mapsto M_2=\{(x,y,z)\in \mathbb{R}^3, x^2+y^2+z^2=1, x<0 \}, \\ \varphi_2(y,z)=(-\sqrt{1-y^2-z^2},y,z).$$
Then I view $x dy \wedge dz$ as the sum of the corresponding two 2-forms on $M_1$ and $M_2$ represented by these two charts.
However this understanding seems a bit cumbersome. I saw a number of similar examples where a differential form on $\mathbb{R}^n$ is introduced but is then integrated over a manifold $M \subset\mathbb{R}^n$. I wonder if there is a simple understanding that I have missed.