I am not sure what's the difference between elisp and emacs-lisp. In my environment, both work for elisp code but I did not found elisp from the load language list:
#+BEGIN_SRC elisp :results output
(print org-babel-load-languages)
#+END_SRC
#+RESULTS:
:
: ((emacs-lisp . t) (python . t) (ipython . t) (plantuml . t) (dot . t) (calc . t) (C . t) (latex . t) (sqlite . t) (java . t) (shell . t))
This works because the function org-babel-execute:elisp exists. Looking at the help for that function shows
org-babel-execute:elisp is an alias for ‘org-babel-execute:emacs-lisp’ in ‘ob-emacs-lisp.el’.
(org-babel-execute:elisp BODY PARAMS)
Execute a block of emacs-lisp code with Babel.
So emacs-lisp is the "real" name and elisp is just a convenient shorthand. You could make more such shortcuts using defalias. For example to have "py" source blocks instead of "python" use
(defalias 'org-babel-execute:py 'org-babel-execute:python)
Elisp.