I believe when most libraries and stores use the term 'ban', they rely on PEN America's definition: "any action taken against a book based on its content and as a result of parent or community challenges, administrative decisions, or in response to direct or threatened action by lawmakers or other governmental officials, that leads to a book being either completely removed from availability to students, or where access to a book is restricted or diminished." [1]

[1] https://pen.org/book-bans/book-bans-frequently-asked-questio...

What would happen if a child brought those not-banned or not-censored books to a library/school where they have "forgo those specific books"? What would the reaction be?

I feel like if they'd still let the person read the book by themselves, and freely share it with others, then indeed it's merely a curation choice. But, if I'd expect, they try to prevent this person from reading their own brought book or sharing it with others, then I think it's fair to say that book been banned and/or censored, at least in that particular location.

No, you need to understand that your specific narrow definition has not handed down by God, and is not more valid than others. US book banning has been a subject for so long now that you are tilling at windmills if you think you can deside what 10000s of people mean when they say banned.

Is a specific institution or library are banned by their decision makers to have a book - that book is banned in that context. If you don't buy this that fair, but don't come at me with your pedantry when I just answered your question.

Yes, this is colloquially referred to as "banning." Sorry, you don't get to decide how others use language.

It is censorship if those books are not included for a specific reason.

“We aren’t including this book in the library because we don’t have space for every book.” <—— not censorship

“We aren’t including this book because we don’t think it’s appropriate for kids to learn about trans people.” <—- censorship