I'm failling to see how a function being globally Lipschitz on a variable in a set is any different from saying that the function has bounded continuous (partial) derivatives on that variable.
Could someone give me some intuition about what means for a function to be Lipschitz?
On
If $f$ is differentiable on an interval in one variable, then $f$ is Lipschitz $1$ on that domain iff $f'$ is bounded there. In several variables the situation is different: For example, $f(x,y) = \arg (x+iy)$ on $\{1< x^2 + y^2<2\} \setminus (-\infty,0],$ where $\arg z$ the principal value argument.
Your intuition is correct...almost everywhere. This is due to Rademacher's theorem. However, as mentioned in the comments, there are plenty of Lipschitz functions that are not differentiable, like $f(x) = \vert x\vert$. In fact, there are even functions with infinitely many nondifferentiable points that are still Lipschitz!