The term "direct flight" seems to change depending who you talk to.
"Non-stop" definitely means from one point to another, without a stop in the middle.
"Direct" sometimes means this, but in many cases it means from one point to another, but effectively with a refueling & dropoff/pickup stop. Generally is the same flight number for both flights, and would be considered a single flight for frequent flyer programs.
To me this isn't really direct, but is still commonly referred to as direct in the industry.
Connecting flights are where it is a different flight number, but still a stop of less than 24 hours. In my book these would be treated as separate sectors, segments, flights, or whatever you want to call them, and therefore should earn points and status credits based on separate flights. After all, you have flown the miles and separate flights, probably in many cases because you didn't have the choice of a Non-Stop flight.