= Solution
Use $S=\operatorname{diag}(1,i)=T^2$ and the <Pauli X gate>. Starting with the <Bell state> $|\Phi^+\rangle$, the exact constructions are
$$
\boxed{|\Psi_1\rangle=(S\otimes X)|\Phi^+\rangle,\qquad |\Psi_2\rangle=(I\otimes H)|\Phi^+\rangle.}
$$
For the first, $I\otimes X$ changes the two terms to $|01\rangle$ and $|10\rangle$, and $S\otimes I$ multiplies just the second term by $i$. For the second,
$$
(I\otimes H)|\Phi^+\rangle=\frac{|0\rangle|+\rangle+|1\rangle|-\rangle}{\sqrt2}=\frac{|00\rangle+|01\rangle+|10\rangle-|11\rangle}{2}.
$$
All factors are single-party <unitary operators>, proving both claimed <local-unitary equivalences>. The equations hold as <vectors>, without even an additional <global phase>.
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