Quick acronym alert:
EDS3 = Ehlers-Danlos Syndrome type III (now know as HEDS)
HMS = Hypermobility Syndrome
HEDS = Hypermobility-type Ehlers-Danlos Syndrome
(B)JHS = (Benign) Joint Hypermobility Syndrome
EDS3 and HEDS are the same thing. BJHS and HMS are the same thing. There have been suggestions that HMS and HEDS are the same thing, (agreed on by many sufferers and some of the most major specialists) but this isn't fully confirmed yet.
Instead of EDS3 you can have EDS1, EDS2, ... all are different types (see the
wiki for more detail). The main reason EDS4 is mentioned in this thread is that it can be important to distinguish between 4 (the vascular type) and 3 (the hypermobility type) as the vascular type can cause sudden rupture of blood vessels etc, so you'd want to know in advance in case something bad happened.
Marfan's Syndrome is another disorder of connective tissue (it affects a protein called fibrillin-1 instead of collagen as with HEDS) which has a wider constellation of common symptoms, so it can be important to diagnose one rather than the others so they know what to fix. You might want to wear a medic-alert bracelet if you were likely to need medical attention and wanted to explain your history quickly - if you often faint, for example. There's a thread on them
hereHopefully that helps...?
Stone