Squared Sub-Gaussian norm of $u^T X$ is uppder bounded by the second moment of $u^T X$

Question

I have read over some papers which use the following assumption of Sub-Gaussian random vector:

(Sub-gaussian observations). The random vector $X \in \mathbb{R}^p$ is sub-gaussian, that is $\|X\|_{\psi_2}<\infty$. In addition, there exist a numerical constant $c_1>0$ such that $$ \mathbb{E}(\langle X, u\rangle)^2 \geq c_1\|\langle X, u\rangle\|_{\psi_2}^2, \forall u \in \mathbb{R}^p . $$

I go back to read Vershynin's ''Introduction to the non-asymptotic analysis of random matrices'', and I found the following equivalent characteriszation of Sub-Gaussian random variable:

1. Tails: $\mathbb{P}\{|X|>t\} \leq \exp \left(1-t^2 / K_1^2\right)$ for all $t \geq 0$;
2. Moments: $\left(\mathbb{E}|X|^p\right)^{1 / p} \leq K_2 \sqrt{p}$ for all $p \geq 1$;
3. Super-exponential moment: $\mathbb{E} \exp \left(X^2 / K_3^2\right) \leq e$;
4. Moreover, if $\mathbb{E} X=0$ then properties $1-3$ are also equivalent to the following one- Moment generating function: $\mathbb{E} \exp (t X) \leq \exp \left(t^2 K_4^2\right)$ for all $t \in \mathbb{R}$.

And the definition of Sub-Gaussian random vector:

(Sub-gaussian random vectors). We say that a random vector $X$ in $\mathbb{R}^n$ is sub-gaussian if the one-dimensional marginals $\langle X, x\rangle$ are sub-gaussian random variables for all $x \in \mathbb{R}^n$. The sub-gaussian norm of $X$ is defined as $$ \|X\|_{\psi_2}=\sup _{x \in S^{n-1}}\|\langle X, x\rangle\|_{\psi_2} $$

However, I still didn't find one that align with the assumption I read in that paper. The second property (Moments) of Sub-Gaussian random variable is similar; however, it is saying the moment is upper bounded by the Sub-Gaussian norm. The assumption I listed on the top is saying the squared Sub-Gaussian norm of $\langle X,u\rangle$ is upper bounded by the second moment of $\langle X,u\rangle$.

May I ask if the assumption on the top applies to any Sub-Gaussian random vector?

I also heard that if $X$ is a random vector, the definition of marginal Sub-Gaussian random vector is different from the definition of joint Sub-Gaussian random vector. So I'm wondering if my question is associated with the distinction of these two definitions. However, I'm not able to get precise definition of these two types of Sub-Gaussian random vector over the internet. Which one of these two alings with the definition of Sub-Gaussian random vector in Vershynin's ''Introduction to the non-asymptotic analysis of random matrices''?