A definition of ‘podcast’ from Dave Winer, who wrote the technical specifications to make podcasting possible:

A podcast is a series of digital media files made available over the open web through an RSS feed with enclosures.

Make no mistake:

If an audio file is not available over the web, or is behind a paywall, or is otherwise exclusive it may be a very fine worthwhile thing, but it is not a podcast. Being accessible openly in a standardized format, RSS, is essential to something being a podcast.