They did not run Lisp as machine code. They either ran a machine code, which Lisp was compiled to, and/or they ran a Lisp interpreter, which ran Lisp source code. The machine often was kind of a stack machine, with microcode implementing the instruction set.
Correct. But for all practical purposes they were lisp machines, you turned them on and got a lisp prompt, not a basic prompt which was the more common language to boot into back then.
* the command interface is based on the Lisp REPL (read eval print loop)
* the operating system itself is written in Lisp
* the instruction set of the CPU is optimized for Lisp (for example via microcode)
* the CPU itself might have hardware support for Lisp (for example for efficient GC)
* the hardware itself (console, keyboard, system bus, extension cards, memory, ...) is used directly from Lisp and might be optimized for it (like special keyboards, special memory cards, special fronted computers, ...)
What's OTOH rarely is the case, that they run Lisp directly in Hardware -> which would mean that the CPU itself is a Lisp interpreter.