Convective overshooting tops reach above the normally horizontal flat layer of the convective system, a layer that should coincide with the tropopause. If we have such an overshooting top, does this mean we actually have the cloud reaching into the stratosphere, or does it mean that the convective system locally pushes the tropopause upward, while the convection is still limited to the (temporarily extended) troposphere? Both alternatives are given as unsourced answers to this identical Yahoo! Answers question, so I would like to see a referenced answer that specifically addresses the distinction between "air parcel reaches stratosphere" or "air parcel pushes tropopause upward".
(Note that this is not merely a matter of terminology, but should affect troposphere-stratosphere interaction in general)