According to the Beit Yoseph, certain sirchos are kosher without any further bedika (examination). These are sircha c'sidran and sircha b'dophen tsar. An animal which has such sirchot is kosher Beit Yoseph, but is not chalak. If it has no sirchos at all, it is chalak. The Rama himself did not allow those sirchos which the Beit Yoseph allowed, in fact the Rama says that wherever a sircha is found, the animal should not be eaten. However he mentions a custom to rub the sircha between the fingers, and if it melts away, it is not considered a sircha at all. The rama does not allow it but says ein mochin b'yadam (meaning we don't stop them). The contemporary poskim all said that the only thing allowed was to rub the sircha between the fingers, but someone who peels off the sircha is feeding treiphos to the c'lal Yisroel. Later, the shochetim started to peel off the sirchos, and for whatever reason, the rabbis allowed it. This is discussed in the Aruch Hashulchan. He specifically says that this peeling off of sirchos that everyone does was formerly considered treiph.
Glatt literally translated means the same as chalak- no sirchos. In practice (it very much depends on the area, the weather, the humidity, the cattle feed, the conditios (free range, small pens, etc.), only about 10% of US beef is truly chalak. To increase the yield, someone decided that if the animal had up to three small sirchos (which they called ririn), it could still be called glatt.
With the demand by knowledgeable consumers for truly glatt (=chalak) meat, the marketers had to come up with a new name. They couldn't call the chalak meat "real glatt" because then they would have to admit that most of the so called glatt really is not. So they decided to call it Beis Yoseph.
So if the meat is called glatt, it most likely is not truly glatt. If it is called Beit Yoseph, it might be truly chalak. If it is called chalak Beit Yoseph, then it is really claiming to have no sirchos.