First, no it wouldn't have been a lute. They weren't invented until the 13th century, if he was playing an instrument it would have been a lyre or cithara; possibly a pandura. None of this matters, however, because my original sentiment still stands. We know Nero wasn't fiddling while Rome burned.
Second,
Also. The article you linked kind of makes it sound like it isn't talking about an instrument at all.
It is the third paragraph.
First it presents the picture of an individual playing a musical instrument as he took a sadistic pleasure in the terrible misfortunes of others. Secondly it represents ineffectual effort.
Pedantry is not a virtue, especially when you're totally incorrect about what you're correcting. I would invite you to read more history.