Are the two notions of subgaussian variable equivalent?

I have seen the following two definitions of subgaussian so far:

Definition 1:Definition 2:

It seems like the only major difference between the two is the constant $2$ in the front. Is there a way to show Definition 1 implies Definition 2? I tried to rescale $v$ to some $v'$ so that $C$ can be adjusted to $2$, but because this rescaling involves $t$, which won't work as $v'$ should be independent of $t$. Any advice on how to derive this?