What does "finger staccato" mean? Is not staccato always done with the fingers?

By no means! There is a well-defined arm staccato, a wrist staccato, and a finger staccato. The latter is produced by a touch similar to the rapid repetition touch—that is, by not allowing the fingers to fall perpendicularly upon the keys, but rather let them make a motion as if you were wiping a spot off the keys with the finger-tips, without the use of the arm, and rapidly pulling them toward the inner hand. The arm should take no part in it whatever.