The questions, despite looking as a representation problem in functional analysis, are much deeper as they bring out the history of the topic involved, notably $BV$-functions and the reasons why the customary definition adopted for the variation of a multivariate function is definition 2 above. And as thus the answers below needs to indulge a bit on this history: said that, let's start.
- If $f$ satisfies definition 1, then do we have $f$ satisfies definition 2? (I felt so but could not prove it rigorously).
No: the two definitions are in general not equivalent. The main problem is that definition 1 is not invariant respect to coordinate changes for all $L^1$ functions: in particular, there exists functions for which the value of the variation $\mathrm{TV}(f)$ depend on the choice of coordinate axes, as shown by by Adams and Clarkson ([1], pp. 726-727) with their counterexample. Precisely, by using the ternary set, they construct a function of two variables such that the total variation according to definition 1 passes from a finite value to an infinite one simply by a rotation of angle ${\pi}/{4}$ of the coordinate axes.
However, for particular classes of functions, the answer is yes: this happens for example for continuous functions, as Leonida Tonelli was well aware of when he introduced definition 1. We'll see something more in the joint answer to the second and third questions.
- If [1] is true then $\operatorname{TV}(f)$ calculated by
definition 1 and definition 2 are they equal?
- If $f$ satisfies definition 2, does there exist a function $g:\mathbb{R}^d \rightarrow \mathbb{R}$ a.e equal to $f$ such that $g$ satisfies definition 1? If so how to prove it?
Since definition 1 is not coordinate invariant in $L^1$ while definition 2 is, for questions 2 and 3 the answer is no. However, things change if, instead of the total (pointwise) variation $\mathcal{TV}$, one considers the essential variation defined as
$$
\newcommand{\eV}{\mathrm{essV}}
\eV(g):=\inf \left\{\mathcal{TV}(v) : g=v\;\; L^1\text{-almost everywhere (a.e.) in }\Bbb R\right\}
$$
(see [2], §3.2, p. 135 or [4], §5.3, p. 227 for an alternative definition involving approximate continuity, closer to the original Lamberto Cesari's approach). Then you have the following theorem
Theorem 5.3.5 ([4], pp. 227-228) Let $f\in L^1_\text{loc}(\mathbb{R}^n)$. Then $f\in BV_\text{loc}(\Bbb R^n)$ if and only if
$$
\int\limits_{R^{n-1}}\eV_i\big(f(x)\big)\,{\mathrm{d}} x_1\cdots{\mathrm{d}}x_{i-1}\cdot {\mathrm{d}}x_{i+1}\cdots {\mathrm{d}}x_n< \infty\quad \forall i=1,\ldots,n
$$
where
- $\eV_i\big(f(x)\big)$ is the essential variation of the one dimensional sections of $f$ along the $i$-axis and
- $R^{n-1}\subset \Bbb R^{n-1}$ is any $(n-1)$-dimensional hypercube.
This result, apart from its intrinsic interest, is valuable since it allows two prove a variation of the sought for result: namely
$$
\sum_{i=1}^n \int\limits_{R^{n-1}}\eV_i\big(f(x)\big)\,{\mathrm{d}} x_1\cdots{\mathrm{d}}x_{i-1}\cdot {\mathrm{d}}x_{i+1}\cdots {\mathrm{d}}x_n =\sup \left\{\,\int\limits_{\mathbb{R}^m}f \operatorname{div}(\phi): \phi \in C_c^1(\mathbb{R^d})^d, \|\phi\|_{L^{\infty}} \leq 1\, \right\}\label{1}\tag{V}
$$
The proof of \eqref{1} follows from the proof of theorem 5.3.5 above in that the method is the same but, instead of the single $i$-th axis ($i=1,\ldots,n$) essential variation, the sum of the $n$ essential variations is considered.
Also, both sides of equation \eqref{1} are lower semicontinuous thus, given any sequence of $BV$ functions $\{f_j\}_{j\in\Bbb N}$ for which they converge to a common (finite) value, it is possible to find a subsequence converging to a $BV$ function $f$: simply stated, the supremum is attained for the limit function of the subsequence and thus it is a maximum.
Thus question 2 and 3 have an affirmative answer if the essential variation is considered instead of the (pointwise) total variation.
Notes
Definition 1 defines the so called "total variation in the sense of Tonelli", and was introduced by Leonida Tonelli only for continuous functions, since the problem of non invariance of the value of variation respect to a change in coordinate axes, pointed out by Adams and Clarkson ([1], pp 726-727), does not exists in this class. The multidimensional total variation defined by using the essential variation, i.e.
$$
\mathrm{TV}(f)=\sum_{i=1}^n \int\limits_{R^{n-1}}\eV_i\big(f(x)\big)\,{\mathrm{d}} x_1\cdots{\mathrm{d}}x_{i-1}\cdot {\mathrm{d}}x_{i+1}\cdots {\mathrm{d}}x_n
$$
is called the "total variation in the sense of Tonelli and Cesari" and was introduced by Lamberto Cesari in [3], pp. 299-300 to overcome the known limitation of definition 1.
I predated reference [1] from the answer by @Piotr Hajlasz to this Q&A: as I pointed out there, definition 1 is the original definition of bounded variation for functions of several variables given by Lamberto Cesari in 1936. Definition 2 was introduced later by Mario Miranda in the early sixties of the 20th century.
References
[1] C. Raymond Adams, James A. Clarkson, "Properties of functions $f(x,y)$
of bounded variation" (English), Transactions of the American Mathematical Society 36, 711-730 (1934), MR1501762, Zbl 0010.19902.
[2] Luigi Ambrosio, Nicola Fusco, Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, New York and Oxford: The Clarendon Press/Oxford University Press, New York, pp. xviii+434 (2000), ISBN 0-19-850245-1, MR1857292, Zbl 0957.49001.
[3] Lamberto Cesari, "Sulle funzioni a variazione limitata" (Italian), Annali della Scuola Normale Superiore, Serie II, 5 (3–4), 299–313 (1936), JFM , MR1556778, Zbl 0014.29605
[4] William P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics, 120. New York: Springer-Verlag, pp. xvi+308, 1989, ISBN: 0-387-97017-7, MR1014685, Zbl 0692.46022
Call $\| f\|_{BV}$ the supremum in definition 1.
Notice that if $f$ is smooth, then
$$
\int_{\mathbb{R}} f(x) \phi'(x) \, dx = - \int_\mathbb{R} f'(x)\phi(x)\, dx,
$$
and so by the duality between $L^1$ and $L^\infty$, we have $\| f\|_{BV}= \| f'\|_1$ (you can absorb the minus sign into $\phi$).
Now, under the same assumption that $f$ is smooth, by the fundamental theorem of calculus,
$$
\int_\mathbb{R}|f(x+h)-f(x)|\, dx = \int_\mathbb{R} \left| \int_0^1 f'(x+th)h\, dt\right| \, dx \leq \int_0^1\int_\mathbb{R} |f'(x+th)|\, dx |h|\, dt = \| f'\|_{1}|h|.
$$
This is the inequality in this case. To pass to general $f$ we mollify.
Consider a function $\eta\in C_c^\infty(\mathbb{R})$ supported on the unit ball, nonnegative, even, and $\int_{\mathbb{R}}\eta(x)\, dx =1$. For $\varepsilon>0$ define $\eta_{\varepsilon}(x)=\varepsilon^{-1}\eta(x/\varepsilon)$ and set $g_\varepsilon= g*\eta_\varepsilon$ for any $g\in L^1_{loc}$, i.e.
$$
g_\varepsilon(x):= \int_{\mathbb{R}} g(y)\eta_{\varepsilon}(x-y)\, dy.
$$
Now fix $\phi\in C_c^\infty(\mathbb{R})$ with $\| \phi\|_\infty\leq 1$, then by Fubini's theorem and using that $\eta_\varepsilon$ is even we have
$$
\int_\mathbb{R} f_\varepsilon(x)\phi'(x)\,dx = \int_\mathbb{R}f(x) (\phi')_\varepsilon(x)\, dx = \int_\mathbb{R} f(x) \phi_\varepsilon'(x)\, dx,
$$
where the last equality is due to the fact that convolution commutes with derivatives. Since $\| \phi_\varepsilon\|_\infty\leq \| \phi\|_\infty\leq 1$, we see that $\| f_\varepsilon\|_{BV}\leq \| f\|_{BV}$. We're now ready to finish: Fix $f\in L^1(\mathbb{R})$ with $\| f\|_{BV}<\infty$ and consider the $f_\varepsilon$ as above, then
$$
\int_{\mathbb{R}}|f_\varepsilon(x+h)-f_\varepsilon(x)|\, dx \leq \| f_\varepsilon\|_{BV}|h| \leq \| f\|_{BV} |h|,
$$
and now simply let $\varepsilon\to 0^+$, using that $f_\varepsilon\to f$ in $L^1$ to conclude.
Added: I should also note that (under the $f\in L^1(\mathbb{R})$ background assumption) the two definition are equivalent, if we pick the right representatives (which is obviously necessary, since elements in Def 1 are equivalence classes, while in Def 2 we talk about pointwise values). This is done in G. Leoni's book "A first course in Sobolev spaces". In your example, the Dirichlet function is a.e. equal to a constant, which is clearly of bounded variation no matter the definition.
Best Answer
The book A First Course in Sobolev Spaces by Giovanni Leoni is a very helpful resource for calculus aspects of function spaces. The first seven chapters deal with functions of one real variable. Chapter 2 is called "Functions of Bounded Pointwise Variation" which are defined by (1) and this class is denoted $BPV(I)$, rather than $BV(I)$. Here $I$ is the interval of definition. The author comments in a footnote on pp.39-40:
Section 7.1 is titled "$BV(\Omega)$ Versus $BPV(\Omega)$". Here, $\Omega$ is an open subset of $\mathbb R$ (we are still in one dimension), and $BV(\Omega)$ is defined as the class of integrable functions $u\in L^1(\Omega)$ for which there is a finite signed Radon measure $\lambda$ such that $\int u\varphi'=-\int \varphi\,d\lambda$ for all $\phi\in C_c^1(\Omega)$. This is somewhat different from, but equivalent to (2): one direction is trivial and the other is a form of Riesz representation.
This is proved thoroughly indeed; the proof takes three pages (pp. 216-218).