Well, believe what you want. What do I know.
The tool you describe as a "Clapper style" is more than likely a spring swage, used predominately with a power hammer, but also used with hand hammering, to form a desired tenon size or to reduce stock size from larger stock. The reason it is split is to allow the piece to be turned while hammering to make it all even.
To form a rivet of any size over 3/16" shank size, you have to have 1) a hole to hold the hot metal, and 2) a rivet header, which forms the extra metal of the parent stock into the domed "head" of the rivet. Rule of thumb for material needed to have enough to form the head is 1.5 times the shank diameter, sticking out above the flat of the header.
Not to begrudge the subject, and this is last I will write on it, but why argue the use of something after being shown pretty good information of what it is????
a quote I read this morning, that I find appropriate right now...
"There are two kinds of beginners: the kind that know everything, and the kind that want to learn everything. First group remain beginners for a long time."
-Colin Davies-