departures are not recorded by DHS. however, DHS can easily obtain the necessary departure records from airlines and border crossing data. thats why there are cases of people who have obtained visas from a consular post at their home countries, and entered the US with the said visas, even though they have previously overstayed their prior visas and/or have lived illegally in the US.
so, while one may lie to get away with favorable dates for time spent outside the US, please be aware that lying to obtain an immigration benefit is a serious issue. And its very easy for DHS to obtain the necessary information.