Actually, in Smash, the move is
キノピオガード
Which is "Kinopio Guard".
So, that would imply that it
is in fact THE Toad and not a generic one.
Or it would....if I didn't find out that "Kinopio" is what is used to refer to
both "the" Toad and the Toad race in general.
http://legendsoflocalization.com/super-mario-bros/names/
The guy that ran this answered a few questions in regard to other Toads (such as Blue/Yellow).
http://legendsoflocalization.com/qa-colored-toads-in-new-super-mario-bros/
http://legendsoflocalization.com/qa-toad-is-trolling-us-all/
(Note, Blue Toad and Yellow Toad are "Ao
Kinopio" and "Kiiro
Kinopio" as opposed to "Ao/Kiiro
Kino".)
This seems to imply that there really is no "the" Toad in Japan, and it's only an American-specific detail formed from outside media.
......that would explain
a lot.