Escrima/eskrima comes from a Spanish word for fencing (related to the word
skirmish; cf. the French word
escrime, fencing). Arnis/arnes comes from a Spansih word for (battle)
harness. The origin of the word kali/kalis is disputed.
Some of the variation is regional (see
this thread); much is choice (Professor Presas' first book was on Modern Eskrima, not Modern Arnis--he later changed it to arnis). The term kali, I understand, is not much used in the Philippines, and other terms, such as Panandata, are used that we hear less of here.
My experience in the States has been that escrima generally refers to arts that emphasize the stick to the (near-)exclusion of other weapons and empty hand; arnis generally refers to an art that includes stick, knife, and empty hand; and kali generally refers to an art that emphasizes weapons over empty hand but uses a wider variety of weapons including swords. You could easily find counter-examples to these generalizations!
Even the three terms you've mentioned leave out many FMAs--Pekiti Tersia, Sikaran (emphasizing kicking), Panandata, and many others.
Opinions on this issue may be quite varied! There are older threads addressing precisely this topic.