This contribution comes with the proviso that it's based on observation of the way my own battered carcass seems to work. Sorry if it doesn't ring true with other people's observations.
So far as I understand, what many people refer to as 'double jointed', and what lots of folks have to some extent (including m'self) have is hypermobility of the joints, caused by too-stretchy ligaments.
This understanding of it is based on a conversation with a 'manipulative therapist' (that's chiropractor by any other name) and reading of a leaflet on the subject while I was working in a hospital.
In my own case, this exhibits itself in the joints (especially elbows and knees) travelling further when extended than they ought. While a 'normal' arm apparently looks straight if extended, mine kind of bends visibly in the wrong direction. This is fine for elbows, as the arm doesn't have to carry as much weight or deal with as much impact as the leg, but my knees are often very painful (especially in cold weather).
I also make clicky-poppy noises from various joints (elbows, knuckles, jaw, neck), some at will, others quite unintentionally, and due to the floppiness of my knee ligaments, find a kind of half-a-lotus-position the most comfortable way to sit, otherwise they kind of set up and are very painful when I stand up.
Anyhow, what I also am lead to believe is that top gymnasts and other people who have to contort to some extent for a living (I've no idea if this includes the sort who fit themselves into tiny boxes at the circus) tend to be "normal-jointed" people who have developed their flexibility through training and practice.
As an example, Mrs 101 is an ex-ballet dancer, now dance teacher, and though she lays no claim to double-jointedness, she can stretch and bend her limbs in a most alarming fashion. She claims this is because she was trained to do it, and likewise, she trains her pupils to do the same thing.
On the other hand, I with my legs which bend the wrong way, my clicky knuckles and my ball-and-socket joint in the right shoulder which I can dislocate at will (Serious. Just the right) also have the tightest hamstrings in the world and cannot even touch my toes, let alone do the splits or anything.
So I think what I'm trying to say is that being"Double-jointed" (whatever we might finally agree it is) is not synonymous with being flexible/ bendy in any practical sense.