Or both?

I think both words work. I would say star in a television show in the past. Then I changed it to star on a television show. If you look on Wikipedia pages, the use is probably 50/50. It comes down to personal preference.

An actor stars on a television show because they appear on multiple episodes over multiple seasons/years. An actor guest stars in/on an episode.

An actor appears on a game show or talk show.

An actor stars in a movie because it is a one time event. It doesn't sound right to say star on a movie.

I think the confusion between in and on stems from acting professionals appearing in different types of shows. For scripted TV and movies, the actors are IN the show as they are usually part of an ensemble. When an actor is invited to a talk show, they are effectively ON stage with the host. For example, Ted Danson may be on The Tonight Show with Johnny Carson while he was in Cheers.

Maybe in has become more widely used because of streaming. It used to be only on television.

There's been a lot of debate about this over the years.

Why is an Actor "In" a Movie, but "On" a TV Show? (https://www.reddit.com/r/NoStupidQuestions/comments/e2d6tz/why_is_an_actor_in_a_movie_but_on_a_tv_show/)

I never thought about it before but yeah I guess it is more common to say an actor is IN a movie and an actor is ON a TV show. Also I sometimes say from for TV actors . Like I'll say Michael Lembeck from One day at a time.

Maybe not the top choice but throwing in a mention of “of”. Valerie Bertinelli of One Day At A Time.

Either/or, why does it matter? LOL.

Ed.