The reasoning behind Rivers regens is ropey at best and probably worth ignoring for its silliness. If making Time Lords was that easy, why not just kick start the race by getting jiggy mid flight? Although, it does sort of help in the argument of whether or not Time Lords and Gallifreyans are one and the same. The reason she can regenerate is because she was conceived in the Time Vortex and so ended up with Time Lord DNA. An improbably story, but if Time Lords were an order and not a race, they wouldn't have their own DNA. My personal opinion though has always been that they are one and the same.
It wasn't just being conceived in the vortex, though. It's said somewhere in AGMGTW that they'd been tinkering with Melody's DNA throughout the pregnancy.
My impression is that the Time Lords are the elite Gallifreyans, not that the two are synonymous, but there's no evidence one way or the other whether they can all regenerate. And the Master beat the limit first by stealing bodies, then by being resurrected (not regenerated) by the High Council.
And yes, I completely agree that what the Doctor told Clyde was probably a joke.
