You know it occurs to me that this assertion was made without being tested. Maybe the elevator wouldn't go up or down at all...
Vision himself says quite explicitly he's not JARVIS.
That aside, in AoU Tony describes JARVIS as starting out as a "natural language user interface" and that "now he runs the Iron Legion and more of the business than anyone besides Pepper."
Indeed, that scene makes it abundantly clear that JARVIS isn't an AI because Bruce and Tony are marvelling at the *possibility* of AI being achievable with the aid of the Sceptre. I mean, you can't marvel at the possibility of something you've already done, no?
In other words JARVIS is a really well made/ programmed SaaS.