You are opening a can of worms for as many different interpretations and opinions as there are people. You make a good observation that people tend to ignore and pooh-pooh as being irrelevant but what is a private business. For example a person working out of their home or having a storefront and living in the back. How about a Lawyers office. I wouldn't call it open to the public exactly like a Wal-Mart but is.
I think the general opinion and there is no real law stating otherwise in most states that signs mean nothing and you have to be told to leave for them to be effective. That is what you are asking and whether this is the way it works in Utah or not I don't know. It varies all over the place and in most cases is actually just opinions.
I know this hasn't answered your question but I really don't think you are going to find an answer to what you are asking.
In SC if a business is properly posted and you carry anyway the store owner does not have to ask you to leave butcan just call the police and have you carted away to jail. If it is not properly posted and he asks you to leave I really don't know if you can refuse or not. The law doesn't exactly cover that.