In several third person (usually limited) stories that I've read, there comes a time when one character shares some of their backstory to one or more other characters. This backstory is always (or almost always) shared with the reader in third person. I tend to find this irritating immersion breaking.
The story is in third person, but I always feel that a backstory told by one character about themselves to other characters would be in first person in the story, and therefore the writer should share this backstory as told by that character, not from the third person narrator.
Question: In a third person story, should a backstory told about a character, by that character be shared with the reader in the first person point of view that the character is giving it in, or would it be better for the author to stick with the previously established third person point of view that the rest of the story is told in?
Note: Please don't use any visual medias in your answers. I get why it would be a bad idea for them to tell a first person story.