"Each such orbital can be occupied by a maximum of two electrons, each with its...

Of course it would be nice if physicists were consistent in their wording, but I don't think this will ever happen. Therefore, my advice is that you don't try to concentrate too much on the definitions like "the spin quantum number is $m_s$", but instead try to understand the concept. Furthermore, this should be simple for you, because I believe that you already got it right:
- Electrons are spin $1/2$ particles. This means that the magnitude of the spin is equal to $1/2$.
- If a particle possesses the spin s, there are $2s+1$ different quantisation configurations: $m_s=\{-s,-s+1,\dots ,+s\}$. These configurations can visualise by assuming that the spin of the particle is only allowed to "point" in certain directions in space (w.r.t. the arbitrarily chosen quantisation axis). So, if $s=1/2$, we get $2\cdot 1/2+1=2$ configurations, $m_s=\pm 1/2$. If instead the spin is $s=3/2$, we get $2\cdot 3/2+1=4$ different configurations, $m_s=\{-3/2,-1/2,+1/2,+3/2\}$.