When random walk is a martingale

I have $$X_1,...,X_n \sim i.i.d.(0,1)$$ and $$S_n=X_1+...+X_n$$,

I define trunctation $$X_i^{(v)}=X_i\textbf1_{\{|X_i|\leq v\}}-E[X_i\textbf1_{\{|X_i|\leq v\}}]$$ and $$S_k^{(v)}=\Sigma_{i=1}^k X_i \textbf1_{\{|X_i|\leq v\}}-E \left[ \Sigma_{i=1}^k X_i \textbf1_{\{|X_i|\leq v\}} \right]$$. So i obtain random walk $$\{ S_i^{(v)} \}_{i=1}^n$$ which increments are bounded by $$2v$$. In my paper there is random walk is mean zero. Thanks to this two properties it's a martingale.

I wonder if there is any definition of martingale I don't know. In my opinion i should have filtration $$F_n = \sigma(X_1,...,X_n)$$ and then checked the definition.

Secondly there is the same conclusion with $$\left\{ S_n - S_n^{(v)} \right\}_{i=1}^n$$. In paper I have it's mean zero P-integrable random walk therefore it's a martingale.

Can someone please explain me or tell definition I need to understand this.

The typical definition of a martingale is that $$\mathbb E[X_{n+1} \mid \mathcal F_n] = X_n$$ where $$X_n$$ is adapted to the filtration $$\mathcal F_n$$. This is equivalent to $$\mathbb E[X_{n+1} - X_n \mid \mathcal F_n] = 0$$ since $$X_n \in \mathcal F_n$$. Note that it is common to take $$\mathcal F_k = \sigma(X_1, \dots, X_k)$$.
Any stochastic process with mean-zero (hence, integrable) increments is indeed a martingale with respect to the natural filtration $$\mathcal F_k = \sigma(X_1, \dots, X_k)$$.