Do something in hook_node_load only when displaying node
Yes, you'll need to examine the URL for this.

Loading a node is (correctly), completely de-coupled from displaying a node. node_load() doesn't know what you've loaded the node for; it doesn't care, its only responsibility is to prepare the node object.

So the context you're looking for doesn't conceptually exist when hook_node_load() is invoked, as that by definition has to happen before the object is passed to node_view(), and there's no way to signal your desired intent to node_load()