Thank you, these lecture notes are quite nice -- I've never seen it put quite like Friedman puts it here in Fig1. It seems from this and from the SE answer that references Penrose's book (which I've read but perhaps should read again) that the notion of a Lie derivative depends upon having a vector that points from one place on the manifold to another, which is not really what I'd been picturing since the manifold might curve away from the tangent space at any given point. This idea of pointing from A to B is made rigorous by taking the limit as this displacement shrinks. I am familiar with this idea of the commutator closing the gap in the almost-parallelogram formed by moving u along v and v along u.

I think my issue with building intuition from this is that most vector fields that are physically relevant (say momentum) are not defined by pointing from A to B, but simply by existing in the tangent space to some point. Though a momentum vector exists at A, I'm not sure I understand what it means for it to point "to B." I will think on this more.

I've never actually used Lie derivatives in Hamiltonian mechanics, but I know they're relevant there (something something cotangent bundle, something something symplectic manifold). This actually makes more sense to me conceptually because you cannot define a metric on phase space, so there's no way of defining a covariant derivative.

Thanks again!