I'm well versed in the oil market. And, yes and no. There are some laws regarding the use of the terminology. But, they do absolutely no good. They give a source the ability to advertise as synthetic without being fully synthetic. Without getting into Group 4 PAO base stocks and such, the synthetic world is so muddled in garbage you really have a hard time knowing what you're buying without looking into the specifications of it.
One of the most used tests quoted recently seems to be the ATSM "Fourt Ball Wear Test"... which is for gear lube and not motor oils... which doesn't make it completely irrelevant, but certainly confuses the playing field even more because the parameters don't mimic an engine.
You can generally tell by cost. A fully synthetic oil will be 3 to 4 times more expensive than a blend. Schaeffer's 9000 line or Amsoil would be examples of full synthetic.
The reason for the motors/transmissions leaking is because of how pure the synthetic oils are. If you have a failing gasket that has been effectively plugged with sludge that has been preventing a leak... then you add detergents to break down sludge and an oil that doesn't have the byproduct of contaminants that create the sludge... you've created a leak. Realistically, your gasket failed a while ago and the problem has just been lurking, waiting for the perfect moment to really flow. (Sat night on the 3-day Monday off weekend in the middle of nowhere, add whatever weather you dislike the most).