Ito’s choice
The Ito integral uses the left endpoint on each interval.
Instead of
$$ \sum_{i=0}^{N-1} Y_{\xi_i}(B_{t_{i+1}}-B_{t_i}), $$Ito chooses
$$ \sum_{i=0}^{N-1} Y_{t_i}(B_{t_{i+1}}-B_{t_i}). $$Why left endpoints matter
The value $Y_{t_i}$ depends only on information available at time $t_i$. In other words, the integrand is adapted.
Because
$$ B_{t_{i+1}}-B_{t_i} $$is independent of $\mathcal F_{t_i}$ and has mean $0$, each increment has conditional mean $0$.
Main benefit
This left-endpoint choice preserves the martingale structure of the integral.