According to my Mrs Beaton, Shepherds pie should properly be made with cold, left-over mutton. Constance Spry has the following to say on the subject of proper cottage pie:

"It is only a modern practice to use cold cooked meat for cottage pie; in days gone by fresh meat was used, finely minced by hand, added to a rich gravy with onions in it, surmounted by a fine mound of the lightest puree of potatoes containing butter and milk, dotted over with butter and crisply browned.

"Now, more commonly, this dish is a rechauffe; even so it need not be a Cinderella among dishes provided it is properly made, well seasoned and flavoured and covered with a really good puree of potatoes properly browned at the last. Unfortunately the potatoes are often too casually prepared and are neither light enough nor sufficiently seasoned or browned."

I have never heard that the ingredients of the two pies are interchangeable.