The answer is both no and yes.
No: S2S chaser points go into S2S total with the S2S activation points (see your S2S log where the 2 components are displayed along with the total). They are scored separately from chaser QSOs.
Yes: An S2S QSO is a special kind of chaser QSO. In a normal chaser QSO, the database doesn’t know or care where the chaser is located but it does know where the activator is because you tell the database. For and S2S QSO, you tell the database where the activator is, then you tell it where you are. It fills in the S2S bits (see above) then enters a chaser QSO in your SEPARATE chaser log. The database is clever enough to know if you have already logged the chase or if you log the chase later how to score everything.
This logging of both the S2S and chase parts in separate parts of your overall log explains why you cannot delete the chaser part of an S2S QSO from the chaser display. If you do you get a red warning message telling you the QSO is an S2S QSO and you need to delete it from the S2S display. It knows that if you have logged both the chase and S2S that you can delete the pure chase.