The $0$th Betti number $b_0$ represents the number of connected components. The $1$st Betti number $b_1$ represents the number of holes and the $2$nd Betti number $b_2$ the number of cavities. While within $3$ dimensions or less, it is clear to me how to derive the $b_0$ and $b_2$, I am struggling with $b_1$ in $3$ dimensions. How do I interpret holes in $3$ dimensions?
In particular having in mind the examples of a ball and a torus in $3$ dimensions. My natural guess would be that the ball has $b_1=0$ which is the correct guess and the torus should have $b_1=1$ because of the hole in the middle. But for the torus it is true that $b_1=2$. If I also count the cavity of the torus as a hole why is that not counted in the case of the ball?
I also read an interpretation of $b_1$ as the number of cuts I can make without disconnecting a component but this interpretation makes even less sense to me when applied to the torus. I guess I can "rip the donut" on one side only. But where else could I rip it without disconnecting the donut?