I want to search for images of this clamp. What do you call this in English?
Binder clip, also known as a foldback or banker's clip.
It is called a binder clip because the bent metal clips may be removed, creating a semi-permanent binding (see right).
This is a Bulldog Clip
so called because it is a brand name registered in 1944.
The type of clip in question is much later and inherited the name in some places and inherited the locally-applied name elsewhere. Sometimes it's called a "fold-back" clip.
I'd suggest the precise type has has a name that varies depending on the audience - and it's a little unclear whether OP is interested in the precise type pictured or the name(s) for all clips that would fulfill this function.
OTOH, it's not a paper-clip.
In Australia, I'm used to them being called Butterfly Clips. Since then, the same name has been attributed to a certain type of hair clips, however.