I'm a native English speaker form the UK. There are problems with all of these examples IMHO.
The use of the simple past "beat" in your first example, followed by "until he was stopped" really grates on my British nerves. It sounds odd in this context. It would be better to use the past continuous "was beating the man until ..."
"Beat on" sounds weird to me in this context. You can beat on a drum, but generally not on a human, well I suppose you technically could but how strange! Ultimately, I don't know if this usage is a colloquial American thing, but it sounds odd to Brits.
"Beat up on" makes no sense to me at all, although it appears in some online dictionaries. It's not used here in the UK. America maybe?
In the UK we usually say "to beat someone up", but this is very informal. In more formal situations it's more likely to hear "The criminal was assaulting the man until the police stopped it", or perhaps "was fighting with the man until ..."
As to whether there is a difference between beating someone up and just beating them, "beating someone up" is an assault, but the meaning of to "beat someone" depends on context. In this context, it's essentially the same. But "to beat" can also mean to win against, as in "He beat me at chess".