In one of the elementary books for stochastic calculus, author didn't clearly explain the difference between $\int(dW_s)^2$ and $\int d(W_s)^2$, using both in the explanations.
Could you please briefly point out the difference between $\int(dW_s)^2$ and $\int d(W_s)^2$?
As I see it: one is squared difference in process , another is difference in squared process
$(dW_s)^2$ does not really make proper sense, but in view of Ito's formula (among some other things) it can be understood as just the same as $ds$. $d(W_s)^2$ means "integration against the process $W_s^2$". You might perhaps understand it as $d(W_s^2-s)+ds$, in which case that first term is now integration against a martingale (which is relatively well-understood).
In terms of Riemann sum approximations your intuition is correct, in an integral with $(dW_s)^2$ you would have $(\Delta W)^2$ in the sum approximations, while in an integral with $d(W_s^2)$ you would have $\Delta(W^2)$ in the sum approximations.