Widespread false formulation of Hunds 1. rule

According to Wikpedia Hunds first rule says

Hunds 1. rule, Wikipedia: For a given electron configuration, the term with maximum multiplicity ($2S+1$) has the lowest energy.

I would formulate this rule more precisely as

Hunds 1. rule, own formulation: The atomic state $|psirangle$ with the lowest energy is an eigenstate of $mathbf{hat{S}}^2$ with maximal total spin $S$
$$mathbf{hat{S}}^2|psirangle=hbar S(S+1)|psirangle$$
Where $$mathbf{hat{S}}=mathbf{hat{s_1}}oplus…oplusmathbf{hat{s_n}}$$
is the total spin operator and $mathbf{hat{s_i}}$ ist the spin operaor of the i-th atom.

In this Oxford Solid State Physics lecture a supposedly equivalent formulation is given

Hunds 1. rule, Oxford formulation: Electron spins align if they can.

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A chemist tried to convince me that the Oxford formulation (usual formulation in chemistry according to him) would be equivalent to the formulation on Wikipedia. But I’m not convinced at all. Let me give a counter example:

Suppose there are just two electrons in the outer shell. $s_1=1/2$ and
$s_2=1/2$. We then know that there are 4 possible total spin states

$$|S=1,m_S=1rangle quad |1,-1rangle quad |1,0rangle quad |0,0rangle$$

According to the Wikipedia formulation of the rule the first three
states are all possibly the ground state. But the second rule only
predicts the first two states to be possible ground states. As can
easily be seen, in the first states both spins are aligned

$$|S=1,m_S=1rangle=|s_1=1/2,s_2=1/2; m_{s_1}=1/2,
> m_{s_2}=1/2rangle$$ $$|1,-1rangle=|1/2,1/2;-1/2,-1/2rangle$$ So
these states are possible ground states in accordance with both
formulations. But the third state does not have both spins aligned,
but instead is in a superposition of anti-aligned states
$$|1,0rangle=alpha|1/2,1/2;-1/2,1/2rangle+beta|1/2,1/2;1/2,-1/2rangle$$ Thereofore it is a possible ground state according to the Wikipedia formulation, but not according to the Oxford formulation. Contradiction.

So the second rule doesn’t make any sense, as it does not account for the existence of superposition states. This is especially bad for more than two electrons, as more and more superposition states arise.

So obviously the second rule is not equivalent to the first. Did I make a mistake or is it indeed a wide spread misunderstanding that the second formulations is equivalent to the first?