How to compute negative point in EC-DSA?
Your second point is the correct interpretation.

Due to the negation map automorphism on (Weierstrass form) Elliptic Curves we have for all affine points that if $P = (x,y)$ belongs to the curve then also $-P = (x,-y)$ belongs to the curve.

This is due the curve's symmetry with the respect to the $x$ axis, as you said.

You can view it as $-k \cdot G = k \cdot (-G)$ but you can also see it as $k\cdot G$ and $-k\cdot G$ are symmetric with respect to the $x$ axis just like $G$ and $-G$.
In fact, how would think of the negation of $(k\cdot G)$ if not with $-(k\cdot G)=(-k)\cdot G$ ?

Now the two point differs. But in the ECDSA verification only the x coordinate is checked, and this will be the same for $k\cdot G$ and $-k\cdot G$