Typically in this nomenclature the first and second parties are the user and OS (or computer) vendor. The third party is an application developer. So typically Google and Mozilla would be third parties the same as maker of the HN browser.
Aside from that, I see no real distinction here. When I launch Firefox, it shows me on my "blank" home page (provided by the browser, not a web page) a list of popular articles curated by its Pocket division. This is essentially identical to what OP's app does. One is a browser that proffers links, the other is a browser that proffers links.
I have seen of lot of funny or meme apps which rip content off same popular websites. I think it unfair to operator of such sites. A lot of these app will show their ads, but won't pull in ads of publisher.
I think this is a bad argument. Showing web content is how the web works.
A third-party application doesn't have the same permissions/protections. Copyright law doesn't always require the author's permission though. If you don't have permission you can still display the content if you have a legal justification. The Fair Use Doctrine is the set of factors to determine whether or not publication without permission is justified.
The transformative nature of the publication is only one factor, although it's the most important. Another factor working in OP's favor is that the use is non-commercial. He says in his blog post that he maintains the site on his own in his spare time, doesn't charge users to download the app, and from what I can tell, doesn't have any ads.
I really think OP could make some small changes and get it approved by the app store.