You specify the timing in the way that results from the optimization problem at hand. The household in your example decides at time t on investment and faces the constraint
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i=k-(1-delta)*k(-1)
i.e. that it has to take the capital stock of last period as given and by investing today decides on the capital stock at the end of the period (i.e. k)
Your second timing would imply that the constraint the household faces at time t only relates to past choices. This constraint would not even involve the investment chosen at time t.