While it was played as a joke, the whole "was she hiding something" had me on first watch going "but everyone is hiding something, surely"

Spock was highlighting this with his trademark precision.

One of the problems that this show has to grapple with is that we already know Spock very well from Leonard Nimoy's portrayal. I'd go so far as to say that - notwithstanding a few gaps - every major event of his life is known to the audience, and we are very familiar with how he's "meant to be".

What then is an actor to bring to such a part? Ethan Peck can't just replicate Nimoy's performance - for one thing, it would be boring. The writers take advantage of this series being a prequel to do the only thing they can: show how Spock became the Spock that we know from TOS. They use his appearance in the TOS pilot "The Cage", where he was visibly more emotional and "human", as a touchstone, and make his journey towards emotional control and "Vulcan-ness" part of his character arc for this show.

I went into it a little in another post, but I think Spock's manner is more familiar and "one of the guys" because he's allowed himself to become emotionally attached to the rest of the crew, and that bleeds into his personality, making him more liable to use humour and jokes to relieve tension. A few years later, when Kirk takes command and many of these officers have moved on, he decides that he will be more emotionally guarded, letting his guard down only with Uhura and Chapel - and only in brief, relaxed moments.

I took as needling the Vulcan Jag…which he couldn’t resist. The set-up was his uncontrolled outburst in the Enterprise mess.