You do not have a "through flight", you have connecting flights (at two different airports - argh!).

I'll disagree mildly with Blumie to extend the definition: Unfortunately, the airlines have used the definition of a through flight being aircraft with the same flight numbers, regardless of whether one changes aircraft or not. It has happened AA had through flight numbers on an M80 DFW-ORD and the same flight number ORD-LHR, as well as a 763 SFO-DFW connecting (sometimes even requiring a terminal change) to another 763 with the same flight number (AA48? iirc) to ZRH.

Regardless of whether it is a through flight or whatever artifact the airlines choose to use, and in your case on connecting flights, you are generally protected if you miss the next segment because of delays, etc. of the first flight - though if the next flight is sold out, I believe you will not have priority over booked, confirmed and present passengers, merely over standby and other flyers, and will go out on the next available flight (and under "Rule 80" might be endorsed to another airline, though Rule 80 is no longer a genuine rule).

And the artifact will often hurt the frequent flyer slightly in miles earning: through flights earn as if they were nonstop flights.