Yes, you are of course right about the different uses of the word ‘school’.
But your original piece, with “One must be taught to think”, implies something other than self-teaching. Your further comment clarifies your thinking, if you will.
Again, I believe we are in agreement.
Thank you for your comments.
Self teaching is typically done with the writings of others as in philosophical or mathematical treatises. In those cases the reader is learning the thought processes of another.
The development of intelligence is a very personal and private activity done in the presence of a greater intellect, whether formally, conversationally or through reading or viewing.
There is "a grappling with another intellect" aspect to the process of developing intellectual skills that can not be avoided.