Number of points of elliptic curve over $\mathbb{F}_p$ with CM by $\sqrt{-2}$ when $p\equiv 1\bmod 8$

Your conjecture is true and follows from a theorem in Cox's book ''Primes of the form $x^{2} + ny^{2}$.'' (Theorem 14.16 on page 317, although this is from the first edition.) In particular, if $\mathcal{O}$ is an order in an imaginary quadratic field, $L$ is the ring class field associated to $\mathcal{O}$ and $E/L$ is an elliptic curve, with good reduction at a degree one prime $\mathfrak{P}$ of $L$, then $\mathcal{O}/\mathfrak{P} \cong \mathbb{F}_{p}$ and $$ |E(\mathbb{F}_{p})| = p+1 - (\pi + \overline{\pi}) $$ where $\pi \in \mathcal{O}$ and $p = \pi \overline{\pi}$. [ This result was originally proven by Deuring, and in fact Deuring's result is more general. ]

Your conjecture follows by taking $\mathcal{O} = \mathbb{Z}[\sqrt{-2}]$, $L = \mathbb{Q}(\sqrt{-2})$ and $p$ to be a prime $\equiv 1 \text{ or 3 } \pmod{8}$ (which implies that $\sqrt{-2} \in \mathbb{F}_{p}$). In this case, your $\pi = a + b \sqrt{-2}$. This theorem doesn't answer the question of what the sign is in the equation $|E(\mathbb{F}_{p})| = p+1 \pm 2a$.

Sorry for self-advertisement: the question of the "sign" that one must choose (equivalently which $\pi$ or $\overline{\pi}$) is entirely answered in my book GTM239, Section 8.5.2, some of the results being due to Mark Watkins (unpublished I believe). The main idea is to show that any CM curve is equivalent in an evident sense to a "basic" CM elliptic curve with a specific equation, giving a relation between the number of points of the initial curve and the "basic" curve, and then Theorem 8.5.8 determines explicitly the number of points of that basic curve. I did not do the exercise for your specific example, but it is completely algorithmic.