He has that introverted extrovert personality but it mainly comes out on stage ("I was born to perform").
I think naturally he is quite shy but as he is also a creative, when it comes to expressing himself, the diva in him sometimes comes out too.

But I think you're right that in the media he's done to promote More, he seems more shy than before.

He might just be a little more self-conscious now as a man in his 60s and aware the perception of the public "Jarvis" is from 30 years ago. He presumably relies less on alcohol and cocaine to get him through social engagements now and he has a grown-up son, he may feel more grounded and content now, and less interest in having to "be Jarvis" and come up with funny or clever quips etc.

Even now has a very endearing trait where he talks quite shyly but then deflects suddenly - through doing an impression/impersonation of an accent or saying something funny which can be disarming in a pleasant way to people not used to his personality.