I found many different definitions of Brownian motion. One is
Def 1: $(B_t)$ is a Brownian motion if it's a.s. continuous, $B_0=0$ a.s., has independents increments and $B_t\sim N(0,t)$.
An other one is
Def 2: $(B_t)$ is a Brownian motion if it's a.s. continuous,$B_0=0$ a.s., $B_t-B_s$ is independent of $(B_u)_{0\leq u\leq s }$ and $B_t\sim N(0,t)$.
I just saw now in a book that
Def 3: $(\Omega ,\mathcal F,(\mathcal F_t),(B_t))$ is a Brownian motion if it's a.s. continuous, $B_0=0$ a.s., $B_t-B_s$ is independent of $\mathcal F_s$ and $B_t\sim N(0,t)$.
So it looks that the last definition is a bit more general and for instance, in the last definition, there is a dependance with a filtration.
Question : Why do we rather consider definition 1 and 2 and not really definition 3 ? Could someone also explain in what definition 3 would be more helpful ?
Definition 3 provides a little more flexibility. For example, if $B_t=(B^{(1)}_t,B^{(2)}_t)$ is a 2-dimensional Brownian motion (2-dim. version of your Def. 1 or 2) and if $\mathcal F_t:=\sigma(B_s: 0\le s\le t)$, then $B^{(1)}$ is a Brownian motion with respect to $(\mathcal F_t)$ in the sense of your Def. 3, but $\sigma(B^{(1)}_t, 0\le s\le t)$ is strictly continaed in $\mathcal F_t$.