The types of Poké Ball available in each game title differ. The Poké Ball type displayed in Pokémon HOME corresponds to the type of ball that was originally used to catch that Pokémon.
If a Pokémon in Pokémon HOME is inside a specific type of Poké ball, and that Pokémon is moved to game data where that type of Poké Ball is not available, the ball will be converted into a regular Poké Ball or Strange Ball depending on the game title.
*Note that if you move your Pokémon back to Pokémon HOME again, the type of Poké Ball will convert back to the original one.