Durrett's definition is the general correct definition of a martingale, while the Wikipedia's definition is at best a "restricted definition". The qualifier "with respect to $\mathcal{F}_n$", although was placed in the parentheses, is essential to accurately define a "martingale": technically speaking, martingale is a sequence of pairs $(X_n, \mathcal{F}_n)$, not of $\{X_n\}$ alone (but the italicized sentence in the quoted paragraph preceding equation $(2)$ below may to some extent justify the latter convention). Patrick Billingsley's book Probability and Measure made this important point more explicit (Section 35):
The sequence $\{\color{red}{(X_n, \mathscr{F}_n)}: n = 1, 2, \ldots\}$ is a martingale if these four conditions hold:
- $\mathscr{F}_n \subset \mathscr{F}_{n + 1}$;
- $X_n$ is measurable $\mathscr{F}_n$;
- $E[|X_n|] < \infty$;
- with probability 1, \begin{align} E[X_{n + 1}|\mathscr{F}_n] = X_n.\tag{1} \end{align}
He continued to explain the role of $\{\mathscr{F}_n\}$ in the definition as follows:
Alternatively, the sequence $X_1, X_2, \ldots$ is said to be a martingale relative to the $\sigma$-fields $\mathscr{F}_1, \mathscr{F}_2, \ldots$. Condition 1 is expressed by saying the $\mathscr{F}_n$ form a filtration and condition 2 by saying $X_n$ are adapted to the filtration.
After that, he illustrated the relationship between Wikipedia's "definition" (which actually is just a special martingale) and the general definition above, which probably can clear out your confusion :
The sequence $X_1, X_2, \ldots$ is defined to be a martingale if it is a martingale relative to some sequence $\mathscr{F}_1, \mathscr{F}_2, \ldots$. In this case, the $\sigma$-fields $\mathscr{G}_n = \sigma(X_1, \ldots, X_n)$ always work: Obviously, $\mathscr{G}_n \subset \mathscr{G}_{n + 1}$ and $X_n$ is measurable $\mathscr{G}_n$, and if $(1)$ holds, then $E[X_{n + 1}|\mathscr{G}_n] = E[E[X_{n + 1}|\mathscr{F}_n]|\mathscr{G}_n] = E[X_n|\mathscr{G}_n] = X_n$ (by tower property of conditional expectation). For these special $\sigma$-fields $\mathscr{G}_n$, $(1)$ reduces to \begin{align} E[X_{n + 1} | X_1, \ldots, X_n] = X_n. \tag{2} \end{align} Since $\sigma(X_1, \ldots, X_n) \subset \mathscr{F}_n$ if and only if $X_n$ is measurable $\mathscr{F}_n$ for each $n$, the $\sigma(X_1, \ldots, X_n)$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale.
Some additional comments in response to specific questions you raised:
- The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
- As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $(X_n, \mathscr{F}_n)$ is a martingale, then $(X_n, \mathscr{G}_n)$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
- At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
You asked for an example, Billingsley also provided many good ones (e.g., Example 35.1 mentioned above), in which "gambling/betting system" is quite illuminating (note that, it is said that the word "martingale" may be originated from gambling, see this interesting vignette for details):
(p. 458) If $X$ represents the fortune of a gambler after the $n$th play and $\mathscr{F}_n$ represents his information about the game at that time, $(1)$ says that his expected fortune after the next play is the same as his present fortune. Thus a martingale represents a fair game, and sums of independent random variables with mean $0$ give one example.
(p. 463) Consider again the gambler whose fortune after the $n$th play is $X_n$ and whose information about the game at that time is represented by the $\sigma$-field $\mathscr{F}_n$. If $\mathscr{F}_n = \sigma(X_1, \ldots, X_n)$, he knows the sequence of his fortunes and nothing else, but $\mathscr{F}_n$ could be larger.
The last sentence "but $\mathscr{F}_n$ could be larger." tells you that, with an example, that Durrett's definition is the correct one and the Wikipedia's definition is clearly not generalized enough. While he did not gave a specific example of a "larger $\mathscr{F}_n$", you can easily conceive some scenarios (e.g., the gambler happens to be an employee of a casino, so in addition to the basic information he should know, he also knows the secret mechanism of the roulette wheel).