This idea originates in homotopy theory and is due to Jardine, "Simplicial presheaves", JPAA **47** Issue 1 (1987) pp 35–87, https://doi.org/10.1016/0022-4049(87)90100-9 ([pdf](https://core.ac.uk/download/pdf/82485559.pdf)).

One can not take for $X$ any site, this definition only works in a topos with "enough points".