Somebody else already called their art Shotokan and Pasaryu. Do you think they made it up on the spot? They were just ripping off what someone else had already taught them and calling it their own work.
I don't know anything about Pasaryu, but this is a very inaccurate description of how the style and name Shotokan evolved. From the Wikipedia entry:
In honour of their sensei, Funakoshi's students created a sign reading shōtō-kan which was placed above the entrance of the hall where Funakoshi taught.http://en.wikipedia.org/wiki/Shotokan#cite_note-Funakoshi1-3 Gichin Funakoshi never gave his style a name, just calling it "karate".
Gichin Funakoshi had trained in both of the popular styles of Okinawan karate of the time: Shorei-ryu and Shorin-ryu. After years of study in both styles, Funakoshi created a simpler style that combined the ideals of the two.[4] He never named his style, however, always referring to it simply as "karate." Funakoshi's karate reflects the changes made in the art by Ankō Itosu, including the Heian/Pinan kata series. Funakoshi changed the names of the kata in an effort to make the "foreign" Okinawan names more palatable to the then-nationalistic Japanese mainland.