I have a question regarding the following two different versions of the Hanson-Wright inequality for estimating the tail behavior of (sub-)Gaussian chaos (i.e. quadratic forms). The first is Thm. 6.2.1 in Vershynin's book "High-dimensional Probability," and it is as follows:
Let $X = (X_1, \dots, X_n) \in \mathbb{R}^n$ be a random vector with independent, mean zero, sub-gaussian coordinates. Let $A$ be a $n \times n$ matrix. Then, for every $t \geq 0$, we have $$ \mathbb{P}(|X^TAX - \mathbb{E}X^TAX|\geq t) \leq 2 \exp\left(-c \min\left\{\frac{t^2}{K^4\|A\|_F^2}, \frac{t}{K^2\|A\|_{op}}\right\}\right),$$ where $K = \max_{i} \|X_i\|_{\phi_2}$. Here $\|\cdot\|_F$ denotes the Frobenius norm and $\|\cdot\|_{op}$ the (spectral) operator norm.
The second I have seen in a couple places, most recently in this blog post, which references Vershynin, and is as follows:
Let $x$ be a random vector with independent, centered $\nu$-subgaussian entries and let $A$ be a square matrix. Then $$ \mathbb{P}(|x^TAx - \mathbb{E}[x^TAx]| \geq t) \leq 2 \exp \left(-\frac{c \cdot t^2}{\nu^2 \|A\|_F^2 + \nu\|A\|_{op} t}\right),$$ where $c > 0$ is a constant not depending on $\nu, x, t,$ or $A$.
My question is, are these two versions equivalent? Both describe the combination of lighter and heavier tails, just using different methods, I guess, but are they interchangeable? I am familiar with the two definitions of sub-gaussian variables which are used here, but I have some trouble comparing them, and I am unsure where the $K$ from the first version or the "$\min$" go... Any help would be appreciated.
I am especially interested if one can recover the minimum version from the sum version while monitoring how the constants change.