= Solution
<Mod-three Galois representation of an elliptic curve> is the representation
$$
\rho_3:G_{\mathbb Q}\longrightarrow\operatorname{GL}(E[3])\simeq\operatorname{GL}_2(\mathbb F_3)
$$
in this case. Twisting by the quadratic character $\chi_d$ replaces it by $\chi_d\rho_3$. Thus $E^d$ has a rational point of order three exactly when $E[3]$ contains a nonzero vector $v$ satisfying
$$
\rho_3(\sigma)v=\chi_d(\sigma)v
$$
for every $\sigma\in G_{\mathbb Q}$: the line $\mathbb F_3v$ is a one-dimensional Galois subrepresentation with character $\chi_d$.
The two-dimensional representation $E[3]$ has at most two distinct one-dimensional characters among its <Jordan–Hölder factors>. Therefore at most two quadratic characters $\chi_d$, and hence at most two rational isomorphism classes of twists, can have a rational point of order three.
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