The other song you asked about is variously known as "The Old Fiddler," "The Old Piper" or, in John Wright's 1970s rendition, "The Old Jews Harper."

There was an old fiddler, old and hoary,
Lived in the town of Ballynorie,
(line not remembered)
And this was all the tune that he could play --

followed by instrumental.

It's in the DT as "There Was An Old Piper."

Bob