The short answer is that the two principal value definitions agree on sufficiently well-behaved functions, but may disagree on sufficiently singular functions. For instance,
on one hand
$$\lim_{\epsilon\searrow 0} \int_{\mathbb{R}\backslash[-\epsilon,\epsilon]} \frac{\mathrm{d}x}{x^3}~=~0$$
is zero, while on the other hand
$$\lim_{\epsilon\searrow 0} \int_{\mathbb{R}} \frac{\mathrm{d}x}{x(x^2+\epsilon^2)}$$
is not well-defined, since the integrand is not integrable at $x=0$.
I) Here we would like to investigate further the definition of principal value $P\int\! \mathrm{d} x$.
Definition. Let $\chi=(\chi_{\epsilon})_{\epsilon>0}$ be a family of functions $\chi_{\epsilon}:\mathbb{R}\to [0,1]\subseteq \mathbb{R}$ that are:
even functions $\chi_{\epsilon}(x)~=~\chi_{\epsilon}(-x),$
Lebesgue measurable functions,
- $\chi_{\epsilon}(x)\nearrow 1$ pointwise almost everywhere for $\epsilon \searrow 0$.
Let us refer to such a function $\chi_{\epsilon}$ as a kernel function.
Examples of kernel functions $\chi_{\epsilon}$ are for instance:
the characteristic function
$$\chi_{\epsilon}(x) ~=~\chi^{\rm std}_{\epsilon}(x) ~:=~ 1_{\mathbb{R}\backslash[-\epsilon,\epsilon]}(x)$$
for the set $\mathbb{R}\backslash[-\epsilon,\epsilon]$. (This choice $\chi^{\rm std}$ will lead to the standard definition of principal value.)
the continuous function
$$\chi_{\epsilon}(x) ~=~ \chi^{a,b}_{\epsilon}(x) ~:=~ \frac{|x|^a}{|x|^a+ \epsilon^b},$$
where $a,b>0$ are two positive constants. (The choice $\chi^{2,2}$ will lead to the other definition of principal value mentioned by OP.)
the constant unit function $\chi_{\epsilon}(x) ~=~ 1$. (Unsurprisingly, this latter choice will turn out to be not so useful.)
II)
Definition. Define the set $V(\chi)$ of $\chi$-admissible functions as
$$V(\chi)~:=~\left\{ f: \mathbb{R} \to \mathbb{C} ~\left|~ \begin{array}{c}
f~\text{is Lebesgue measurable},\cr
\forall \epsilon>0~:~~ \chi_{\epsilon} f~\in~ {\cal L}^1(\mathbb{R}),
\cr \text{and} \cr
\left(\int\! \mathrm{d}x~ \chi_{\epsilon}(x) f(x)\right)_{\epsilon>0} \text{is convergent for}~ \epsilon\searrow 0 \end{array}\right.\right\}. $$
Definition. If a function $f\in V(\chi)$ is $\chi$-admissible, we define the $\chi$-based principal value as
$$P(\chi)\int\! \mathrm{d} x f(x)~:=~\lim_{\epsilon\searrow 0} \int\! \mathrm{d}x~ \chi_{\epsilon}(x) f(x).$$
Here ${\cal L}^1(\mathbb{R})$ denotes the set of functions that are Lebesgue integrable, i.e. functions that are Lebesgue measurable and whose absolute value has a finite integral. ${\cal L}^1(\mathbb{R})$ is an example of an ${\cal L}^p$ space.
III) It is not hard to see that:
If $f\in{\cal L}^1(\mathbb{R})$ is Lebesgue integrable, then it is $\chi$-admissible $f\in V(\chi)$, and the principal value
$$P(\chi)\int\! \mathrm{d} x ~f(x)~=~ \int\! \mathrm{d} x ~f(x)$$
is just the ordinary Lebesgue integral because of the
Lebesgue dominated convergence theorem.
The set $V(\chi)$ of $\chi$-admissible functions is a $\mathbb{C}$-vector space.
If a function $f\in V(\chi)$ is $\chi$-admissible, so is the mirrored function $(x\mapsto f(-x))\in V(\chi)$, with same principal value.
If an $\chi$-admissible function $f\in V(\chi)$ is odd, then $P(\chi)\int\! \mathrm{d} x~f(x) ~=~ 0$.
Thus it is enough to investigate even and odd functions.
Finally, let us investigate power functions $x\mapsto x^p$, $p\in\mathbb{R}$, which play an important role in practice as building blocks.
IV) Even functions. Let
$$g_{p,K}(x) ~:=~ 1_{[-K,K]}(x) |x|^p~=~g_{p,K}(-x)$$
be a truncated power function, where $p\in\mathbb{R}$ is a real power, and where $K>0$ is a positive truncation constant.
It is not hard to show that in the case of Example 1, 2, or 3,
$$g_{p,K}\in V(\chi) \qquad \Leftrightarrow \qquad p>-1\qquad \Leftrightarrow \qquad g_{p,K}\in {\cal L}^1(\mathbb{R}).$$
In the affirmative case $p>-1$, the principal value definitions based on the three Examples 1, 2, and 3 agree:
$$P(\chi)\int\! \mathrm{d} x ~g_{p,K}~=~\int\! \mathrm{d} x ~g_{p,K}~=~ \frac{2K^{p+1}}{p+1}.$$
V) Odd functions. Let
$$h_{p,K}(x) ~:=~ {\rm sgn}(x) 1_{[-K,K]}(x) |x|^p~=~-h_{p,K}(-x)$$
be a truncated power function, where $p\in\mathbb{R}$ is a real power, and where $K>0$ is a positive truncation constant. In the three Examples 1, 2, and 3, we get
- $h_{p,K}\in V(\chi^{\rm std})$ always,
- $h_{p,K}\in V(\chi^{a,b}) \qquad \Leftrightarrow \qquad p+a>-1$,
- $h_{p,K}\in V(1) \qquad \Leftrightarrow \qquad p>-1\qquad \Leftrightarrow \qquad h_{p,K}\in {\cal L}^1(\mathbb{R}).$
