Two definitions of “semistable locus” for GIT quotient

I am reading Nick Proudfoot's notes on geometric invariant theory and projective toric varieties in which he gives a definition of "semistable" which is unfamiliar to me but which is apparently equivalent to the usual one.

Let $G$ be a reductive algebraic group acting on a projective variety $X$, and let $\mathcal{L}$ be a linearization of this action. For $x\in X$, we say that $x$ is semistable if there exists $m>0$ and a $G$-invariant section of $\mathcal{L}^{\otimes m}$ whose non-vanishing locus is affine and contains $x$. This is the most common definition I have encountered, but it is rather opaque to me (and I would be very happy if someone were able to provide some intuition for why this is a reasonable condition).

The definition Proudfoot gives is more geometric: he says that $x$ is semistable if for all nonzero $\ell\in \mathcal{L}^*_x$, the closure of the $G$-orbit through $(x,\ell)$ in the total space of $\mathcal{L}^*$ is disjoint from the zero section. This definition feels somewhat more natural to me, but less computationally workable than the above. Is there an easy way to see the equivalence of these two definitions?

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