At a technical level, a podcast is an audio or video show distributed via an RSS or Atom feed through feed enclosures.
A podcast client (previously known as podcatchers) is therefore an application that allows users to subscribe to said RSS or Atom feeds and download those media enclosures when a new one is made available.
In order to be more approachable to users who have no idea how the technical underpinnings of podcasts work, most modern-day podcast clients have on-board directories. The act of having a directory in itself is not an issue, as long as the ability to add arbitrary RSS feeds via a URL is preserved. This ensures that podcasting remains an environment where users can bring their own client, and no single player in the market can attempt a hostile takeover of the podcasting space.
YouTube shows are not podcasts if they only live on YouTube, as you can't subscribe to them in a podcast client. Spotify exclusive "podcasts" are not podcasts. While the underlying technology still uses RSS feeds, those RSS feeds are inaccesible outside of Spotify and violates the principle that users should be able to choose which app they wish to consume podcasts in.
Paid podcasts are not all bad though. Podcasts such as Dithering, which are locked behind a subscription paywall, still do so in a way that is friendly to any podcast app the listener decides to throw at it. It is specifically the act of gating "podcasts" behind a specific app that runs counter to the openness podcasts were designed around.
Sometimes we are asked why Limitless Possibility isn't available through some particular app, and generally the answer is that the app they are asking about is actively working against these principles, and we choose to withhold our podcast from directories that do so. If the app was well-behaved to begin with, you could simply copy and paste the RSS feed URL into your desired app and subscribe without any intervention from us.