"Upon a time when Burbage played Richard III, there was a citizen grown so far in liking with him, that before she went from the play she appointed him to come that night unto her by the name of Richard the Third. Shakespeare overhearing their conclusion went before, was entertained and at his game ere Burbage came. Then message being brought that Richard the Third was at the door, Shakespeare caused return to be made that William the Conqueror was before Richard the Third. Shakespeare's name William. (Mr Tonse)"What can we conclusively prove from this? Not much, but it does suggest a lot of things. I hypothesize this joke was told to Manningham by "Mr Tonse," presumably a fellow student, hence the citation at the end. It's consequently unknown from whence Mr Tonse heard it, or if he invented it. If he did invent it, good on him. If not, it was a joke that was going around, so it suggests that Shakespeare and Burbage had sufficient celebrity status to merit being the subject of such jokes. There's little to suggest that this is an account of an actual event, no more so than a joke about any celebrity or politician pecadillos in Dave Letterman's opening monologue. But it could suggest that Shakes and Burby had a reputation for being Elizabethan mack daddies.
I also get a kick out of how Manningham feels compelled to explain the joke at the end, suggesting to me that although he appreciated a good joke, he wasn't all that good at delivery. Skimming through the diary (hooray for YahooBooks), it seems that all the clever or funny bits are cited from colleagues.
Even if it doesn't tell us anything definitive about Shakespeare or Burbage, it's still a funny joke.