I would like to ask about the definition of a simple normal crossing divisor. Let me take the definition for instance in Kollar's book Lectures on resolution of singularities.
Let $k$ be a field (one may assume $k=\mathbb{C}$) and $X/k$ a smooth variety. Let $D \subset X$ be a divisor, we say it is a simple normal crossing one if $D$ is smooth and every of its irreducible components is smooth and all intersections are transverse. That is, for every $p \in X$, there exists a regular system of parameters $(x_1,...,x_n)$ of $\mathcal{O}_{X,p}$ (being a regular local ring since $X$ smooth) and natural numbers $(m_1,...,m_n)$ such that $D$ is cut out by $\prod_{i_j \in I}x_{i_j}^{m_{i_j}}=0$ for some subset $I \subset \left \{1,...,n \right \}$. ($m_i$'s are called multiplicities).
This is of course equivalent to requiring that for every $p \in X$, there exists a neighborhood $U$ of $p$ and global sections $(x_1,...,x_n) \in \Gamma(U,\mathcal{O}_X)^n$ so that their restrictions to local rings about $p$ form a system of local coordinates and $D=V\left(\prod_{i_j\in I}x_{i_j}^{m_{i_j}} \right)$ is cut out by the same equation as above.
Let me take a concrete example: if $k =\mathbb{C}$, and $X = \mathbb{A}_{\mathbb{C}}^2$, $D = \left \{y^2=x^3+x+1\right \}$ (one can check that $D$ is smooth by the Jacobian criteria). Is $D$ a simple normal crossing divisor? In particular, can we say that $(X,D)$ is a resolution of singularities of itself? As far as I understand, since $D$ is smooth, it is regular, codimensional $1$ in $X$ so that locally, $f_1=y^2-x^3-x-1$ can be viewed as an element of a local coordinate with two elements. Globally, it is impossible, so how can one find (locally) the other coordinate $f_2$ matching with $y^2-x^3-x-1$ (hence $D$ is identified with $V(f_1)$) to form a local coordinate $(f_1,f_2)$?
I think the points of the definition of a simple-normal-crossing (snc) divisor $D=\sum_i a_iD_i\geq 0$ are the following. (i) each component $D_i$ is smooth, (ii) at each point $P\in D$, the local defining equation of $D$ is of the form $x_1^{b_1}\cdots x_m^{b_m}=0$ for suitable local coordinates $x_1,\dots,x_m,\dots,x_n$ at $P$ where $n=\dim X$ and suitable positive integers $b_i$.
For example, in the affine plane, $D=(xy=0)$ is a snc divisor but $D=(xy(x+y)=0)$ is not a snc divisor. Another example is that $D=(xy(x+y-1)=0)$ is a snc divisor and $D=(y^2=x^3+x^2)$ is not a snc divisor. So, in your sentences of definition, $D$ itself is not necessarily smooth. I think other sentences are correctly stated.
For your concrete example, $D=(f_1=0)$ is a snc divisor (I didn't check the smoothness, though). $(X,D)$ is a log-resolution of itself. Finally, the system of parameters can be constructed as follows. If $P=(a,b)\in D$, take the local coordinates $x-a,y-b$ which originates $P$. The equation of $D$ becomes $$f_1=\frac{\partial f_1}{\partial x}(a,b)(x-a)+\frac{\partial f_1}{\partial y}(a,b)(y-b)+(\text{higher terms in }x-a,y-b)$$ by the Taylor expansion, while the smoothness of $D$ at $P$ translates to $(\frac{\partial f_1}{\partial x}(a,b),\frac{\partial f_1}{\partial y}(a,b))\neq (0,0)$. By the Jacobian criterion, $x-a, f_1$ is a system of parameters near $P$ as long as $\frac{\partial f_1}{\partial y}\neq 0$, and $y-b, f_1$ is a system of parameters near $P$ as long as $\frac{\partial f_1}{\partial x}\neq 0$.