How common is instantaneous capital production

In macro growth models, consider the following versions of the capital evolution: $k$ is capital, $d$ is a depreciation rate of the capital, and $i$ is an investment. $’$ represents the next period.

$$k’=(1-d)k+i$$
$$k’=(1-d)k+i’$$

The first one is the typical capital accumulation evolution. The second one is different in that the available capital tomorrow is concurrently produced.

My Question:

(1) How common is the second evolution, and if you have seen it, in which model types have you seen?

(2) Understanding the difference: the way I understand the difference is that the first one refers to a situation where the production happens so late in the period that by the end of that time interval, it is so close to the next period, the produced capital is carried over to the following period. In the second version, I think of a situation where the production happens so early that the produced capital is available within the same period. Is my understanding correct?

(3) If so, is this distinction possibly for pure mathematical convenience in which the model employs such evolution?