That's the very reason I ask for the story or the history behind the song.
I agree with you but I still think you are confused the song itself with the title of the song. Are we talking about the name Lao Damnern Sai, or the melody? And what is the point of this kratoo or topic - the meaning of the name, right?
I still believe the meaning lies in the story behind the song. I vaguely remember the story behind it as I said...about a stroll along a sandy beach, but I may be wrong. You seem to know a lot about Traditional Thai music, so, can you check it out for me?
The name given to this Lao-flavoured song is obviously Thai, regardless of how the author got inspired - by copying an existing Lao song, by adjusting the Lao song or songs, or by thinking it all up, or whatever.
And in return for your agreeing to check on the history of the song, I can gladly check for Laotian language point. I don't know much about Laotian, but from what I know I think I can tell you your hypothesis is not quite likely. The Laotian are simple and easygoing people, their language is a purer, more innocent version of our own language, not so "rung-rang" with Khmer, Pali, English, etc, as our Thai. They say "yang" to mean dern, and that's that. You said maybe to glorify the word as to be suitable for a title of a song, a word "damnern" was substituted. That is exactly the Thai thinking, not the Laotian. But I will double-check.
I also agree with you that the English translation sounds strange. I just follow your logic, that if you compare Lao Damnern Sai with Lao Damnern Kwian, when the one means (your own translation) travelling by the cart, the other must, by logic, mean "travelling by the sand (

!?)" which is, of course, strange. I wanted to point that out. Your proposed translation, travelling on the sand, or Dern Pai Tam Sai, is almost exactly my own translation - a stroll along the sand (beach- as I still think of the beach on the river as part of the story I heard.) I am glad you seem to agree with me on my translation.
From my little knowledge of Laotian, I don't think the Laotians would say - Dern Pai Tam Din Tam Sai, not likely. But I will check.
The point about the foreign-derived melodies being Thai or not- that's beside the point of the topic which is just about the meaning of this particular song's title. But it is an interesting point in itself and I will be happy to discuss it in another kratoo.
Sorry I can't write in Thai now.