Check if an Elliptic Curve point (over finite field) is nearer than another point

It's unusual to consider distance between points on a discrete Elliptic Curve as used in cryptography. Several different distances can be defined. Which one do you want, or what is the actual purpose?

Example making some sense: 1) The number of times we must use $G$ to go between points $G^a$ and $G^b$, that is $|G^a,G^b|=|((a-b+\frac{n-1}2)\bmod n)-\frac{n-1}2|$ where $n$ is the order of $G$. 2) "Manhattan/taxicab" distance per Cartesian coordinates, for points other than the point at infinity, that is $|A,B|$ is the integer $|((x_A-x_B+\frac{p-1}2)\bmod p)-\frac{p-1}2|+|((y_A-y_B+\frac{p-1}2)\bmod p)-\frac{p-1}2|$, where $p$ is the field order. 3) "Flying" distance $\sqrt{(((x_A-x_B+\frac{p-1}2)\bmod p)-\frac{p-1}2)^2+(((y_A-y_B+\frac{p-1}2)\bmod p)-\frac{p-1}2)^2}$

If by _"private key"_ it's meant what's 9857, 54 and 448 in the question, then one can compute the distance as in 1 above (then compare) only when that distance is below some threshold (e.g. $2^{72}$), or the private keys of the points involved are known. For distances 2/3, only the points are needed (but these other distances do not seem so useful).

I sit in a Tesla and translated this thread with Ai: