It could. That might normally be considered cheating but LISP is actually designed with philosophy of customizing language itself to problem domain. Thing is, the implementations and even language features of modern LISP's are nothing like McCarthy's simplistic language and interpreter. They couldn't succeed being that due to performance and usability issues.

So, I think it's more fair to talk about LISP languages as an ongoing family but still a cheat to treat a modern one as original LISP. Probably should date it from creation of dialect or similar one.