You are correct. Here is the relevant WOJ I found:
Should Harry’s first deal with Lea been covered by Maggie’s deal?
A: Not really. One problem is that Harry dealt with Lea the way that one would deal with another human, so he really didn’t get much out of the bargain besides some confidence. Lea gave him the “magic feather” so to speak.
And, Maggie could have done a little better job on her deal with Lea, but she was kind of in a rush, so she wasn’t as detailed as she should have been.
I am thinking that Harry has been played by the Winter Court since his conception. Margaret saw the need for a star child and agreed to conceive on in return for his protection by the Winter Court, hence Harry has a real fairy godmother in the form of Lea.. The Winter Court gets not only a star child but an eventual winter knight.. I think Harry's fate has always been more or less directed by others, but not so much to over ride his free will, he still has choices. Yeah, Jim compared Harry to Dumbo, as you will remember Dumbo could always fly, but he didn't believe it. Young Harry didn't believe he could beat Justin though he had the power all of the time.. Lea's mumbo jumbo gave him the confidence.. However I don't think the bargain he made with her was very binding, the real contract was made between Margaret and Mab long before he was born, he just doesn't fully realize that yet.