Where It Sits, Where It Happens に・で

書く and 置く both end with something landing at a fixed spot, the name ends up on the notebook, the present ends up here, which is exactly what に marks, the destination point of the result, not the setting of an ongoing activity.

する here describes an activity actively unfolding, playing tennis, doing homework, and で always marks the stage that activity plays out on. This is the same で you already know from by-at-with, extended to mark any location where an action, not just a static state, takes place.

いる and ある, existence verbs, almost always take に, どこにいますか, never どこでいますか, since existing somewhere is a state, not an action. Any verb describing an actual activity, however, takes で, even inside the exact same coffee shop.