Is Bingo the farmer or the dog? It's a genuine question. On the internet. Here's how we'd answer it. The first line of the rhyme is:
There was a farmer who had a dog,
And Bingo was his name-o.
Now, if there was an extra comma, like this:
There was a farmer, who had a dog,
And Bingo was his name-o,
then we'd agree that the farmer was called Bingo.
And, just to clinch our argument, the first recorded transcription read:
The farmer's dog leapt over the stile,
His name was little Bingo.