What are the fundamental differences between citrus fruits that necessitate different cooking techniques?
There are three (to the best of my knowledge) main factors at work here:

Pectin.  This is a gelling agent, a bit like gelatin.  Fruits such as apples and plums are very high in pectin; citrus fruits are not that high in pectin but citrus peels are incredibly high, so if you're using any of the peel, you're getting tons of great pectin.
Obviously, the more pectin, the faster and firmer it's going to gel.  If the fruit doesn't contain much pectin then you're going to have to cook it down a lot or just add actual pectin (you can buy it in stores).

Acidity, as you suspect, makes a big difference.  It's not just acidity, it's the relationship of acidity to pectin as above.  For more details read The Role of Acid in Jelly Formation.  Basically, pectin gels best at a pH of 2.5 (almost exactly the pH of lemon juice) to 4.5 (about the pH of sweet grapes).
It's best to hit the sweet spot in the middle rather than the extreme ends, so you won't generally be using pure lemon juice, you'll be adding some sugar and other ingredients.  Which leads me to the final factor:

Sugar.  As described in The Role of Sugar in Jelly Formation, sugar precipitates the pectin, basically making the pectin molecules less soluble which mimics the effect of a higher concentration.  Therefore, more sugar makes the jelly set faster.


So: Pectin-rich fruits + high acidity + high sugar content = fast setting.
There are actually various other factors that affect the pectin (which is pretty much the only gelling agent at work when working with natural fruit jellies) like salts and even alcohol, but in the vast majority of recipes, it's acidity and sugar that are your primary concerns as well as the actual pectin content of the fruit itself.