It is my view that no matter how many times a silly line has been re-printed, it's still silly. Old-time cowboys probably talked pretty much the way we do, and their songs ought to make sense to us.

Thus the line

'I ride an old paint and I lead an old dan...'

is probably bogus. Nobody knows of a noun 'dan' which has anything to do with cowboys.

It is reasonable to suppose that Paint is the name of one horse and Dan is the name of another. From what I have read, cowboys were not given to fanciful names for their mounts, and a painted horse called Paint is likely.

That changes the first line to

I ride old Paint and I lead (2 notes on 'lead') old Dan (2 notes)

It evokes a clear picture and it keeps the listener from being distracted from the goal - which is to share the world of a cowboy.