There is an option to play with career averages, rather than individual player-seasons, so I guess a "career stats draft" is just a draft using that option.
From the description on "Diamond Legends Career Open League" (when you're picking the type of league you want to join):
"Select this option to draft for a league with player career numbers adjusted to fit into a 162 game season, instead of individual player seasons.
If you're looking for a league where there can only be one instance of a player, you can join/create a theme league using a league rule (league-wide restrictions) and a live player draft (no clones)."