Which capital accumulation is right? $K_t = (1-\delta)K_{t-1} +I_t$ or $K_t = (1-\delta)K_{t-1}+I_{t-1}$? In a lot of models with capital, I find different variants of capital accumulation formula as follows:
$K_t = (1-\delta)K_{t-1} +I_t$ or 

$K_t = (1-\delta)K_{t-1}+I_{t-1}$

Which one is more economics-sound formula?
Both are economically sound. The notation is just a question of convention. The reason behind the ambiguity is that capital is a stock and investment is a flow variable. You are looking at capital in two different instants. Investment happens during the time between the two instants and its index is either the starting or the ending instant.