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:
> 1. $\mathscr{F}_n \subset \mathscr{F}_{n + 1}$;
> 2. $X_n$ is measurable $\mathscr{F}_n$;
> 3. $E[|X_n|] < \infty$;
> 4. 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: 

1. 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 $(X_1, \ldots, X_n)$ is 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 $(2)$. 
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.  
3. 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](https://www.jehps.net/juin2009/Mansuy.pdf) 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).