Elizabeth, you can always be counted on to cut thru the BS and get to the right answer.
I agree about Tatum. I have his solo Masterpieces set, and I don't play them very much. After a while they all sound the same. Show off.
Peterson is not always great, but he is at least always very good. Sort of like clint eastwood in the movies.
Basie's style of playing in Jazz is sort of like B.B. King's style of playing the blues. The right note, at the right time, no more or less. You nailed it.