Jon - in the pieces that I mentioned in the other thread ("Pernod" by Johnny Cunningham and Michael O'Domhnaill, "Plum Creek" by Grey Larsen, and "Viimeinen Valssi" or "The Last Finnish Waltz"), each has a tune as you describe it. However, I don't feel that it's descriptive to call the entire piece a "tune", because there is so much more there than just the melody, or tune.
One of the reasons that I found myself wondering about this is because when I looked at the other submissions, I saw that they were all (as far as I could tell), songs with words. I wondered if maybe I was broadening the definition of the word "song" by applying it to pieces of music that were purely instrumental.
Irish sergeant and McGrath of Harlow - maybe you've got it right. Maybe it's all in the ear and heart of the beholder.
Liz the Squeak - I think I can get behind you on that whalesqueak thing.