OK. So, if I'm reading this correctly, your issue is the following:
\frac{1}{\sqrt{2}}(|\uparrow\downarrow> +|\downarrow\uparrow>)
corresponds to a total spin of 1, while
\frac{1}{\sqrt{2}}(|\uparrow\downarrow> -|\downarrow\uparrow>)
corresponds to a total spin expectation value of 0.
The easiest way to see that this is indeed the case is to explicitly work out the expectation value for the total spin operator:
\hat{S^2}=\hat{S_1}^2+\hat{S_2^2}
You should explicitly see that for the first state the expectation value is s=1, while for the second, s=0.
Another, possibly more physical way to think about the problem is the following:
Consider the Singlet State:
Apply the space rotation operator to the state, and you'll notice that it is invariant under all rotations. Only systems with 0 angular momentum can be rotationally invariant, so we know it's spin must be 0.
Similarly, the m=0 triplet state will not be rotationally invariant, so we know it has some angular momentum.
Thus, even though the states "look" similar, they have very different rotational properties, thus different spin values.
