It is in general $\textit{not}$ true that $\textit{if}$ $A$ and $B$ are elliptic curves over the finite field $K$, lying in the same $K$-isogeny class, and with the same $j$-invariant, $\textit{then}$ $A$ and $B$ are $K$-isomorphic (see below). However, this is true if $A$ and $B$ are ordinary. You can find a related observation in Waterhouse's thesis (Abelian varieties over finite fields, Éc. Nor. Sup. 1969, remark page 542). Let me describe a different approach, similar to that suggested by Silverman.
I tried to be succinct in my answer, I failed as there are several details to verify. Sorry about this.
Let me slightly reformulate the question. Let $K$ be a finite field of size $q$ and characteristic $p\geq 5$. Let $E$ an elliptic curve over $K$, denote by $\pi_E:E\to E$ the Frobenius isogeny of $E$ relative to $K$. By Honda-Tate theory, $\pi_E$ defines a Weil $q$-number whose minimal polynomial over $\mathbf{Q}$ dentifies uniquely the $K$-isogeny class of $E$. Let $E'$ be a $K$-form of $E$, with associated Weil number $\pi_{E'}$. Your first question can then can be formulated as:
Can we have that $\pi_{E'}$ is conjugate to $\pi_E$ without $E'$ being $K$-isomorphic to $E$?
(Here by "conjugate" I mean that the $\pi_E$ and $\pi_{E'}$ have the same minimal polynomial over $\mathbf{Q}$.)
To answer the question let's start by describing the $K$-forms of $E$ in a slightly different (but equivalent!) way than that used by Silverman above.
Set $W_K=\textrm{Aut}_K(E)$ and $W_{\bar K}=\textrm{Aut}_{\bar K}(E\times_K\bar K)$. Since $p\geq 5$, $W_{\bar K}$ is isomorphic, as a $G_K$-module, to $\mu_n$, where $n=2,4$ or $6$. Moreover, we can think of $W_{\bar K}$ as the roots of unity inside some imaginary quadratic field $F$ embedded in $\textrm{End}_{\bar K}(E\times_K\bar K)$, where $F$ is unique if $E$ is ordinary or if $n>2$.
The set of $K$-forms of $E$ is in bijection with the (group) $H:=H^1(G_K,W_{\bar K})$. Consider the two cases
i) $W_K=W_{\bar K}$;
ii) $W_K\subsetneq W_{\bar K}$.
In the first case the evaluation map $c\mapsto c(\varphi)$ of $1$-cocycles on the arithmetic Frobenius $\varphi\in G_K$ induces an isomorphism $H\simeq W_{\bar K}$.
In the second case the action of $G_K$ on $W_{\bar K}$ is by inversion, and the evaluation map $c\mapsto c(\varphi)$ as above induces an isomorphism $H\simeq W_{\bar K}/(W_{\bar K})^2$.
You can check that this description of $H$ matches the Kummer-theoretic one given above by Silverman (at least as abstract groups).
Now, if $c$ is a $1$-cocyle (for $G_K$ acting on $W_{\bar K}$), then Silverman in his book explains how to construct an elliptic curve $E^c$ over $K$ $\textit{and}$ an isomorphism $E^c\times_K\bar K\simeq E\times_K \bar K$.
One can then show (I owe the omitted proof to Jakob Stix) that in the ring $\textrm{End}_{\bar K}(E\times_K\bar K)$ the following relation holds
$\pi_{E^c}=c(\varphi)\pi_E$,
where we make an implicit use of the identification $\textrm{End}_{\bar K}(E\times_K\bar K)\simeq \textrm{End}_{\bar K}(E^c\times_K\bar K)$ induced by the isomorphism $E^c\times_K\bar K\simeq E\times_K \bar K$ encoded by $c$.
In other words, the Weil numbers $\pi_{E^c}$ and $\pi_E$ differ by multiplication by a root of unity. If the cohomology class defined by $c$ is non-trivial, the above formula implies that the non-trivial twist $E^c$ and $E$ are $K$-isogenous if and only if
$\bar\pi_E=-\pi_E$,
where $\bar\pi_E=q/\pi_E$ is the complex conjugate of $\pi_E$. Notice that from what we saw above $c$ is non-trivial in $H$ iff [$c(\varphi)\neq 1$ in case i), and $c(\varphi)\not\in W_{\bar K}^2$ in case ii)].
This last condition on $\pi_E$ then amounts to require it being purely imaginary, hence conjugate to $\sqrt {-q}$. If $q=p^e$, then $\sqrt{-q}$ arises as $\pi_E$ for some elliptic curve $E$ iff [either $e$ is odd, or else $e$ is even and $p\not\equiv 1$ mod $4$]. (In the remaining cases where $e$ is even you get $K$-simple non-geometrically simple surfaces attached to $\sqrt{-q}$).
The end of the story is that if $e$ is odd or $e$ is even and $p\not\equiv 1$ mod $4$, then every elliptic curve $E$ in the $K$-isogeny class defined by $\sqrt{-q}$ admits $|H|$-many $K$-isogenous, pairwise non-isomorphic $K$-forms. All these are examples of supersingular elliptic curves ($p$ divides the trace of $\sqrt{-q}$) with not all geometric endomorphisms defined over $K$. They exhaust the instances of the phenomenon we were after.
