I have been working through Vershynin's "High-Dimensional Probability" recently, and have been having some trouble with the chapter on subgaussian random variables (2.5). In Proposition 2.5.2, one of the equivalent definitions of a sugaussian random variable is the following condition on the moment-generating function (MGF): $$\mathbb{E}\left[e^{\lambda^2 X^2}\right] \leq e^{K^2_3 \lambda^2},$$ where $|\lambda| \leq \frac{1}{K_3}$, and $K_3$ is a constant. Later, after introducing the subgaussian norm, this condition is restated the following way: $$\mathbb{E}\left[e^{\lambda X}\right] \leq e^{C \lambda^2 \|X\|_{\psi_2}^2}.$$ when $X$ is a zero-mean random variable and for some (new) constant $C > 0$.
Now, I don't quite understand how we get from one to the other. I would guess it is not too hard, since it was just given as restating the proposition, but I think I haven't quite understood the transition from the constants in the proposition to the use of the norm. I thought perhaps each constant was just equal to some scalar multiple of the norm, but I can't explain to myself why we need zero-mean in this case, or why the squares disappear.