I don't think there is really a most general sense; rather there are various different senses.
"Calculus" is Latin for "pebble". The "-ul-" suffix indicates that it refers to something small. In dentistry, "calculus" means mineral deposits in people's teeth. Among modern senses of the word, maybe that one is closest to the original meaning. The word "calculate" ultimately comes from the use of pebbles in abaci for calculating. Various modern senses come from that. When the word is used for the differential and integral calculus and also for the predicate calculus, they're two different meanings that evolved from the sense of calculating, not two instances of a general meaning. Some things like "vector calculus" and "calculus of finite differences" evolved from the sense in which the word is used when it refers to differential and integral calculus.
Addendum several minutes later:
See Calculus (disambiguation).