I've read that anecdote as well, but looking at that "inverted" earlier illustration, I just can't fathom how that would flip to give us the orientation we know today.
Consider, the reasonable spot to secure a string would be upon the thicker cylinder (what is now the engineering hull) towards the end aligned with the saucer. Tell how THAT could flip to have the cylinder underneath and the saucer on top! Somebody activate a grav-plate mounted upon the celing during that fateful meeting? (Sorry, that was a bit snarky.)
Now, what I imagine could have happened is this. The wooden model is revealed, the longer cylinders simply resting upon the desk or table. Roddenberry lifts it and slowly turns it around in his hand, staring at it from various angles. Finally, he turns it over with the thick short cylinder now underneath. Since he stated the ship never lands upon planets (due to the effects budget needed to depict this on a weekly basis), he realizes this orientation better implies this thing CAN'T land, that it is a craft built purely for space travel.
I have my suspicions this is just another of those colorful anecdotes that makes for more interesting stories than the boring reality.