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Are the two notions of subgaussian variable equivalent?

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

Definition 1:enter image description hereDefinition 2:enter image description here

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?


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