the terminology of door check and door closer are from long
ago,
in 1880's and 1890's door closers were just a spring device to
pull the door shut but not really control the closing speed or
force,
door check, that was more of a hydraulic or pneumatic cylinder
with a piston inside (think of a regular screen door closer, but
without the spring) was used together with a door spring to create
some controlled door closing, and prevention od door slamming.
now, the term of door check is faded because after the rack and
pinion hydraulic closer was invented, the dampening effect was
added onto the device in the manufacturing process. hence now they
are all referred to as door CLOSER.
hope this helps, expert in closers here