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 (by tower property of conditional expectation) $E[X_{n + 1}|\mathscr{G}_n] = E[E[X_{n + 1}|\mathscr{F}_n|\mathscr{G}_n] = E[X_n|\mathscr{G}_n] = X_n$. 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.
In short, $\sigma(X_1, \ldots, X_n)$ is the smallest filtration $\mathscr{F}_n$ that you can have with respect to which $\{X_n\}$ is a martingale. In words, the "past history" should not be interpreted as the sequence of interested random variables only, it may cover any information up to time $n$, which is in general much larger than $\{X_n\}$ itself.
You asked for an example, Billingsley also provided a good one (note that, it is said the origin of the word "martingale" is 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).