When did "Grover Cleveland" become Pete?

Growing up it was always Grover Cleveland Alexander. When did he become Pete? Anyone else remember when it was "Clarence" Gaston? How about "Joey" Belle?

Looks like Bill Killefer, his longtime battery mate in Philly, called him Alkali Pete because his appearance on a Texas hunting trip was similar to a couple of silent movie cowboys, on named Alkali Ike and another named Mustang Pete.
Kinda morphed into Ol' Pete.............

In Belle's case it was by his own request.

I too remember Grover Cleveland Alexander being more common growing up, in fact I don't think I ever heard Pete until Baseball Reference came online. BR generally just copied the names listed in Macmillan's Baseball Encyclopedia, but I just cracked the 8th edition of Macmillan's and they list him as Grover Alexander. (Imagine, I had to go find a book to look something up, having flashbacks to the 1980s now.) So my theory is we can blame BR, but now I'm curious why they diverged from Macmillan's in this instance.