Let $E/Q$ be an elliptic curve given by a Weierstrass equation $E : y^2 = x^3 + Ax + B$ with $A, B ∈ Z$ and,
$h_x(P) = \log H(x(P))$, where, $x(P) = p/q, H(x(P))=\max\{|p|, |q|\}$.
There is a constant $C_2$ that depends on $A$ and $B$ such that $h_x([2]P) \geq 4h_x(P) − C_2 $
for all $P ∈ E(Q)$
But in $E : y^2 = x^3 + 1$ we see that, a point $P = (2, 3)$ is on $E$, and $2P = (0, 1)$ also on $E$, these two pints do not follow the equation $h_x([2]P) \geq 4h_x(P) − C_2$, for those two pints the equation becomes- $0 \geq 4\log (2) − C_2$.
Why is that? How do we calculate $C_2$?
Why do you say that the points do not follow the "equation" (inequality)? For any $C_2 \geq 4 \operatorname{log}(2)$ your inequality holds; what is the issue?
By the way, notice that the real substance of the claim is that we can find a single constant $C_2$ such that $h_x([2]P) \geq 4h_x(P) − C_2$ for all $P$. As in the example above, if you only look at a fixed $P$, it is trivial that such a $C_2$ exists.
As for finding such a $C_2$ explicitly, it may not be so nice. If you haven't already, you can look at the proof of this claim in Section VIII.3 of Silverman's book The Arithmetic of Elliptic Curves. Silverman does not attempt to keep track of the constant explicitly but by carefully going through the steps of this proof it may be possible to produce an explicit value that works.