Notation for vector fields on manifolds

Question

I am a bit confused with the way Jack Lee (and my teacher in class too) defines vector fields. For instance, in Introduction to Smooth Manifolds, page 182, Example 8.17, Lee defines a smooth vector field on $\mathbb{R}^{2}$ by $$Y = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}.$$ Is this just shorthand for $$\forall (x,y) \in \mathbb{R}^{2} \, \colon \: Y_{(x,y)} = x \frac{\partial}{\partial y}\Big\vert_{(x,y)} - y \frac{\partial}{\partial x}\Big\vert_{(x,y)},$$ or am I missing something, i.e., should the $x$ and $y$ multiplying the vector fields $\partial / \partial y$ and $\partial / \partial x$ be interpreted as functions here?