From the programmer's perspective C is not a stack based language the way that, say, Forth is, because (as you note) there are no explicit push and pop instructions. But from an implementor's perspective, evaluating C function calls and arithmetic expressions is very stack oriented, which is why in the cited paper some Bell Labs researchers in the early 1980s were trying to build stack machine CPUs to execute C code. Given your Lisp experience, just think about how you'd translate various C expressions to Lisp ones, and how Lisp expressions map to stack operations, and you'll see C's stack based nature come out. (I.e. the abstract syntax trees (ASTs) a C compiler builds can naturally be represented as Lisp-like expressions.)

C:

  y = a*x + foo(b);

Lisp:

  (setq y (+ (* a x) (foo b)))

Example stack instructions for the above (many variations possible):

  PUSH a
  PUSH x
  MUL
  PUSH b
  PUSH foo
  CALL
  ADD
  PUSH y
  SETQ  ; or MOV or whatever your arch wants to call it

Dennis Ritchie added the register storage qualifier keyword to primeval C (and also auto, inherited from B) to make the earliest C compilers easier to write, because bug-free register allocation is a very hard problem and Ritchie's earliest PDP had only a few kilobytes of core to hold both the C compiler's code and the chunk of program text being currently translated, so a complicated register allocator was out of the question.