I actually thought the idea of a conversation as the exchange back and forth between two people would be well accepted in email clients, but having looked around it seems all email clients that refer to a 'conversation' actually display the thread view, i.e. emails linked together on a common subject regardless of who they're from. The Microsoft Outlook documentation in particular refers to the ability to view a "Conversation or Thread" which looked like you could have either when in fact Microsoft is using both "conversation" and "thread" to refer to the same view, grouping emails by their subject line.
This conflation of two different ideas will probably hamper the development of a more contact-centric view of email for another decade.
So in summary,
a thread view would be with an inbox grouped together by subject line, and
a conversation is the list of emails exchanged with a given user, e.g. as iPhone messaging: