JC's Tune Finder is a search engine that searches all the abc tune file database sites on the internet that he knows about. As such it's a lot like Google in the sheer amount of data it references, much of it probably duplicates.

The reason for different table widths is that data is output in groups of 20 and the column widths are dynamic, i.e., they change to accommodate the amount of data in them. The two columns at the far right are the culprits that usually make a table wider, since they can vary quite a bit in size. Thus a table of 20 items will be sized to accommodate the width of the widest element in any given column.

I suppose one way of winnowing the list would be to look at the far left column (if you chose the default "Single" in the search options; if you chose "Multiple" it'll be the second column on the left). You'll note very few of the matching tunes have an entry in that column. This entry refers to the source of the data, usually one of the well known tune books, such as, in the case of The Blarney Pilgrim, O'Neill's or The Fiddlers Companion.

These might or might not be the versions your teacher uses. I find my teacher tends to teach tunes as he is currently performing them, which may vary quite a bit from ANY source material. In some cases, he hasn't looked at the sheet music for them in years. Since I record my lessons I usually find it easiest to just transcribe his playing of the tune into NoteWorthy Composer.