Measuring multi-qubit state with respect to an arbitrary basis

Suppose Alice and Bob each hold qubits; they have a joint state

$$|psirangle = frac{1}{sqrt 3}big(|00rangle + |01rangle + |11ranglebig).$$

Alice measures the first qubit in some basis (say $|+rangle, |-rangle$); I want to see what Bob’s qubit looks like post-measurement. The result should be a a probabilistic ensemble (mixed state). How do we do this in practice?

I thought about it this way: measurement in a basis is equivalent to applying an appropriate unitary matrix before measurement. So for $|+rangle, |-rangle$, we’d apply

begin{align*}
(H otimes I)|psirangle &= frac{1}{sqrt 3}bigg(frac{1}{sqrt{2}}(|0rangle + |1rangle)|0rangle + frac{1}{sqrt{2}}(|0rangle + |1rangle)|1rangle + frac{1}{sqrt{2}}(|0rangle – |1rangle)|1ranglebigg)
&= frac{1}{sqrt 6}bigg(|0rangle|0rangle + |1rangle|0rangle + 2|0rangle|1ranglebigg)
end{align*}

But when we measure the first qubit, we get "$|0rangle$" (really $|+rangle$) with probability $frac 16 + frac{2^2}{6} = frac 56$, leaving the state as $frac{frac{1}{sqrt 6}big(|0rangle|0rangle + 2|0rangle|1ranglebig)}{sqrt{frac 16 +frac 46}}$ (is this correct to say?). Likewise we could get "$|1rangle$" (really $|-rangle$) with probability $frac 16$, leaving the state as $frac{frac{1}{sqrt 6}|1rangle|0rangle}{sqrt{frac 16}}$.

This would mean that post-measurement, Bob’s qubit has a state described by the density matrix

$$rho = frac 56frac{|0rangle + 2|1rangle}{sqrt 5}frac{langle0| + 2 langle 1|}{sqrt 5} + frac 16|0rangle langle 0|$$

Does this make sense? In particular, does the $H otimes I$ operation make sense?

I ask because when I take $operatorname{tr}_A |psirangle langle psi|$ I get $$rho_B = left( begin{matrix} frac{1}{3} & frac{1}{3} frac{1}{3} & frac{2}{3} end{matrix} right)$$ which seems to be a different density matrix altogether.