I definitely also say "Apple TV app", but I'm just not convinced there officially is such a service! (Not sure if that's more a statement about me or Apple's marketing/branding department.)

There used to be iTunes, where you could buy and rent movies. Then Apple split that app into its components "Music" (including purchased music as well as the subscription service called Apple Music), "TV" (including movie rentals, purchases, and the subscription service "Apple TV+") and "Podcasts".

Since these three things are quite generically named, of course it makes sense to practically refer to them as "Apple Podcasts" etc., and Apple's marketing department should have seen that need coming, but in-ecosystem, it does align somewhat nicely with other apps called Calendar, Mail etc, and when it's not clear from context, just add "Apple" in front (and maybe "app" at the end), and everybody will know what you're talking about.