Why am I getting the wrong result when applying the extra strong Lucas pseudoprime test?
You're missing the $\pm$ on the $2$ in the first condition.
A number $n$ passes the test if one of the following conditions holds:
- $U_d \equiv 0 \pmod n$ and $V_d \equiv \pm 2 \pmod n$.
- $V_{d \cdot 2^r} \equiv 0 \pmod n$ for some $r$, $0 \le r < s$.
In your case, $U_7 \equiv 0 \pmod{13}$ and $V_7 \equiv 11 \equiv -2 \pmod{13}$, so the first condition holds.