Clip type for in-system programming (SOIC8 vs DIP8)
You are correct, the pin pitch is different so you'd need a dedicated DIP clip or a dedicated SOIC clip. I think the description for that product may be misleading.

EDIT: The chip is socketed so you can carefully pry it up and program it outside of the circuit, like on a breadboard. This is probably what's best unless there's a dedicated header on the board for in-circuit programming. Taking it out of the circuit removes ambiguity as you can control what it's connected to. You won't need a test clip anymore either.