Joel & Joseph are interchangable names;

I think it is only a (poor) transcription that has resulted in Joseph rather than Joel; although it may be that when the record was originally recorded "Joseph" was entered. Joel & Joseph are distinct christian names.

1861
Joseh Wait abt 1831

is a reasonable transcription Josel or Joael would have been acceptable too, in fact the late Joael is probably as close phonetically as you will get (?)