The Lisp 1 Programmer's Manual is dated 1960 and ... bears McCarthy's name.
The paper you cited has a section titled The LISP Programming System which says:
"The LISP programming system is a system for using the IBM 704 computer to compute with symbolic information in the form of S-expressions."
The Programmer's Manual's Preface begins:
"LISP 1 is a programming system for the IBM 704 for computing with symbolic expressions."
Looks like they are about exactly the same thing to me: an implementation for a specific computer (IBM 704), described in the same year by the same author, with an a nearly identical sentence in the intro.
http://jmc.stanford.edu/articles/lisp/lisp.pdf
Lisp was always as much as an idea as a real proramming language and both were developed together. Lisp was developed as a practical tool for AI research, especially for his AI research.
"One mathematical consideration that influenced LISP was to express programs as applicative expressions built up from variables and constants using functions. I considered it important to make these expressions obey the usual mathematical laws allowing replacement of expressions by expressions giving the same value. The motive was to allow proofs of properties of programs using ordinary mathematical methods. This is only possible to the extent that side-effects can be avoided. Unfortunately, side-effects are often a great convenience when computational efficiency is important, and “functions” with side-effects are present in LISP."
Reading above, in his original idea, he clearly had the idea of mathematical pureness for LISP, but for practical reasons the real implementation became one with side-effects.