I think you need to read Masato's post, and mine, again. 1656 is the date of the second edition. The first, by definition, would be earlier. The British Library makes an educated guess at 1655, but it isn't something you need to worry about. I explained that the book is a collection of separate pieces, the names of which Masato has quoted. Singing Simpkin is just one of them, that's all; and whether or not there is a connection with Will Kemp has nothing directly to do with Staines Morris. I mentioned it, and him, because the suggestion that one piece in the collection might have been based on an earlier stage routine may be taken as presenting the possibility that others were, as well.

Since I take it that none of us have actually seen the book (my reference was bibliographical) we don't actually know whether Chappell meant that the song in question was from the piece Diana and Actæon, or from the book of that title, the other contents of which are detailed above.

That being the case, and in the absence of other information, the most likely assumption is the obvious one; that Cox wrote the words to the song. It's a fairly typical pastoral lyric of the sort popular on the stage of the day. We don't know what tune it was originally sung to; it was some 200 years later that Chappell set it to the Staines Morris tune. That tune is the right age to have been used originally, but so are a great many more.