Interested in words, I had to peek in the unabridged OED.
It says the words, L. educat-educare, to rear or bring up children, and L. educere, to lead forth, are related. Nothing is said about one coming from the other.
Bald Eagle, you are right in that words receive new meanings as our use of words changes through time. The following quotes from the OED suggest that the modern meaning came in at least by the 16-17th centuries.
1607- "The ... horses are not to be despised, if they were well bred and educated." 1628- "Himselfe, delighting in the Rivers and Mountains, among which he had been educated." 1588- Do you not educate youth at the Charg-house at the top of the Mountains?" 1540- "The education of infantes at this tyme ..."
Educere leads to our word educe.
1669- Chaos was that ancient slime, out of which al things were educed." 1678- "All life...is... educible out of nothing and reducible to nothing again."
This shows that the words had separate meanings by the 17th century. I do not have the references or knowledge of what the meanings were in ancient Rome, or in medieval times. Our understanding of Latin is much colored by renaissance scholars.