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In the tools that embed the "Rename" option (Select, Append Fields, Join, Join Multiple), copying the new name will copy all the information of the field configuration : tick/untick, original field name, type, size, new name and description.
Renaming the field "Rename_Field"
In my opinion, it should copy only the new name. This would be useful, especially because when you change the name of a field, it isn't automatically changed in subsequent tools, so copying it to replace it in those tools is faster than retyping it every time.