See the attached graphic for clarification. Here I have created four tasks 1,2,4 and 5 in a scheduling tool.
Task 1 has a FS relationship with Task 2
Task 4 has a FS – 3 days relationship with Task 5
As you can see, both of the relationships are FS. However the later has a lead of 3 days (i.e., - 3 days).
Your definition, "Activity must finish before successor can start", holds for a strict FS relationship. A FS relationship with a lead in the successor is not a strict FS relationship; it’s an adjustment of the FS relationship.
Does this help?
Regards,
Khurram