Wouldn't a ships AI be different than the synths and positronic brains?
In all likelihood they are.
The vessel in question could easily be run by adaptive algorithms to execute commands in line with SF rules and regulation (and heavily insulated to protect from outside breaches/hacking).. but not 'true AI'.
Today, AI CAN be used to describe a wider range of software, but in Trek, there seems to be a distinction where the designation of 'AI' would mean a self aware synthetic lifeform (but not say adaptive algorithms by themselves).
Synthetic lifeform, artificial intelligence, Photonic (sapient hologram) - I'd guess that the Federation placed an insta-ban on any inorganic intelligence within their borders. I mentioned elsewhere, I could imagine Starfleet officers going from door to door on every world, using specialized phasers to exterminate any synthetic being on sight. The Zhat Vash's stupid little pawns.
And yet, Rios used what are effectively sentient holograms onboard his ship inside UFP space... holograms are AI too, just in a different form.
No. The AI ban likely extended mainly to AI's inside cybernetic bodies such as say androids who have greater degree of 'freedom' than your regular holos.
The Doctor's holomatrix was adaptive by default and programmed to adjust/learn but he was still a hologram. It was kinda inevitable that he would effectively develop sentience from that alone... but he's not an android... and was declared a person a long time before the AI ban was in place (its possible some exceptions would have been in place for Data had he survived and the EMH).
And he already proved his loyalty to SF and the UFP at large. And I doubt the rest of the VOY crew would have allowed for his program to be terminated under the AI ban.
If there was an 'AI purge' (which I doubt), the simplest thing would have been to just take those synthetic lifeforms offline without destroying the underlying technology in which their software resides. That way, you don't 'murder' a sentien AI (bad or not)... you just put it into stasis.
Regular holograms by themeselves aren't adaptive. Some, such as the ones RIOS used likely had limited self-awareness in the sense that they are holos, but otherwise, they didn't seem particularly 'adaptive' in behavioral aspect like the doctor... some had adaptiveness in terms of fighting styles... but if they lack the same for their behavioral subroutines, then they couldn't really develop sentience.