Homemade press

[Toolmaker51]

"Die" is a correct term; applied on tools that cut or form in conjunction with a punch, called a "die set".
Sometimes a distinction is made, naming them according to function, especially "forming die". Dies that cut are a bit more common, and "die" is sufficient.
While a metal brake forms by clamping and bending, a press style exhibits a more distinct entry of a punch into a die.
Large dies performing a series of operations, are "progressive". Usually fed from long rolls of material, not necessarily just flat stock, according to finished part.
'Drawing' is a forming operation too, common parts are door knobs, drawer handles, firearm cartridges...The stock can be sheet or thick slugs.
There are dies doing a set of operations, often hand fed from station to station, of individual pieces. These work in many instances; such as long parts with each end being trimmed in a radius. One I built, made a metal reinforcement for truck mud flaps. Upper edge had long bevels left and right, lower side had two radius's. It also stamped the manufacturer name in the middle. That company didn't have a wide press with sufficient tonnage for a single hit, the build accommodated that. Two hits, cut one bevel and radius left then right, stamping occurred once with the right side.
The subject of types of dies go far deeper than this!