Closure plate is an occasionally used secondary term for a Register plate - its not a secondary meaning, or a different kind of plate with a different function but merely another infrequently used alternative term for a Register plate.
Now that google has muddied the waters with claims for different names and plate functions its become confusing for those that are not experienced in these things.
Register plate has been the most consistent term that i've heard.
All Register plates were and are made from metal - even when called a closure plate the plate must still be metal: nothing else.
In the UK ree a closure plate is different from a register plate. A register plate is used when a metal liner isn't used and the original flue takes the gases.
It takes the stove pipe and must be totally sealed to stop any escape of fumes and metal to protect it from breakage in the event of a falling brick etc.
A closure plate is used when a s/s flexible liner takes the gases and is more to stop any soot falling down. It can be a cement board which makes for easier cutting, although it must be non combustible.