No, STARZ is a network.
When they say it's "a STARZ original series" they mean it's original on their network. As in, not a rerun from any network or a new series bought from another network. A network original. It's marketing doubletalk.
Yes, I understand what you said. I still question the use of the word "original" in favour of "First time on STARZ".
"Original" does not imply "not a rerun" to me. It implies "We invented this".