Why is inserting in the middle of a linked list O(1)?

You are correct, the article considers "Indexing" as a separate operation. So insertion is itself O(1), but getting to that middle node is O(n).

The insertion itself is O(1). Node finding is O(n).

No, when you decide that you want to insert, it's assumed you are already in the middle of iterating through the list.

Operations on Linked Lists are often done in such a way that they aren't really treated as a generic "list", but as a collection of nodes--think of the node itself as the iterator for your main loop. So as you're poking through the list you notice as part of your business logic that a new node needs to be added (or an old one deleted) and you do so. You may add 50 nodes in a single iteration and each of those nodes is just O(1) the time to unlink two adjacent nodes and insert your new one.