I agree. I would say intelligence is what you have naturally, education is what you acquire in school, and wisdom is what you acquire through experience. These are not interchangeable.
By three methods we may learn wisdom: First, by reflection, which is noblest; Second, by imitation, which is easiest; and third by experience, which is the bitterest.
Yes, familiarity with a field should improve skill, but I like to think that one can be smart, intelligent or wise in a situation that does not involve previous information about it.
Someone exposed to a brand new game or puzzle can be smart, intelligent or wise in solving it separately (to some degree) from having prior information.