It had existed since the fall of 1981, in fact — and kept the name for 15 years!
With its launch on September 7, 1996, Fox Sports Arizona and ASPN (or Cox Sports — Dimension had been sold at this point) began sharing a channel on the Cox cable system. Cox Sports held the rights to the Suns, but Fox Sports had Pac-10 football (including its opening night telecast of an ASU football game and quite a few games in the Sun Devils' Rose Bowl season, though it was not until 1998 that the university moved its own games from KTVK) and the newly relocated Coyotes.
If you want to be more confused, ASPN apparently picked up Prime Sports programming. (Prime Sports' LA feed was even available for a time in Tucson!) It wasn't until June that Prime wanted to set up its own Arizona channel:
"Prime move? Prime Sports, the subsidiary of Fox-Liberty Sports, is negotiating with Cox about leaving ASPN for a separate channel of its own. It would house the Diamondbacks (in two years), along with with Coyotes and its Pac-10 football and basketball packages. If a separate channel is landed, Cox may pay as much as a 50-cent-per-subscriber increase to Prime, which, in turn, undoubtedly will be passed on to subscribers in the monthly basic cable cost. Johnson says negotiations are ongoing, but said a decision hasn't been reached and ASPN remains a viable outlet."
In between that (which turned out to be what happened) and September 1, the Prime Sports brand was dropped for Fox Sports. While it was announced that the rest of the RSNs would not change until October, the new Arizona network was exclusively referred to Fox Sports Arizona in the paper, though apparently it did use Prime Sports Arizona for two months.
The Suns remained with Cox for some time after. For the 1998-99 NBA season, Cox Sports was replaced with the more general-themed Cox 9 (later Cox 7, then YurView Arizona — the rebrand also allowed Cox to give KAZT channel 7 on cable). Fox Sports picked up the Suns at some point after in the early 2000s. YurView continues to dabble in sports besides high school, particularly with GCU, though it has also telecast Arizona Rattlers and other smaller sports events in the past.