I've seen a couple of references to intellgence as a shorthand for education, and I'm not sure I agree with that assessment -- after all, many people have access to education, and the relative success or failure that any individual accomplishes with their own education seems to me to have as much or more to do with factors that have little to do with intelligence (for instance: persistence, which is a big reason for the difference in my first college experience, where I washed out with a sub-2.0 GPA, and my second, where I graduated with honors).
If anything, education seems like more a thing that happens to you, and in that sense it seems better expressed in game mechanics as...experience? (This suggests to me that the Third Edition mechanic of skill ranks that eventually dwarfed your stat modifier in a skill is more 'realistic' than the Fifth Edition mechanic where your best stats will generally be near your proficiency bonus. After all, how often do you run into someone who is really good in a specific field, but seems hopeless in any other area of life? That seems to happen to me a lot.)
I see intelligence as more of a trait that allows a character to perform abstract thinking, adapt to new situations, find connections or patterns, or otherwise apply existing knowledge or understanding to novel situations. 'Quick-thinking' would be part of it, but not everyone who is a quick thinker is necessarily intelligent to the same degree.
As a DM, I like asking for Intelligence checks to allow characters to recall some seemingly-obscure bit of information that nevertheless has some application or provokes an insight useful to the current situation. The application of this approach in fighting monsters (monster lore) or dealing with puzzles seems obvious, but it can also be helpful in giving a character a chance to provide the Help action with a non-intellectual puzzle -- the smart character can recall a story about a farmer taken to the Faerie King and relate some salient details to the rogue, who is trying to persuade the local nobleman to spare his life for seducing the noble's daughter, as an example.
Playing a character smarter than you is, in that sense, no different than playing a character stronger or more dextrous than you -- just figure out what kinds of problems the character might be able to use his intelligence to solve or help solve, and let the dice fall where they may. (Honestly, I find it much more difficult to play a character with significantly higher Wisdom than my own.)
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Pauper