Because "Branding" is a word with strong connotations/meanings, it can confuse a discussion. Would you be okay using terms such as expectation and perception, which I think cover what you're talking about, rather than branding which tends to imply the visual design identity.
Actually I would prefer other people understand what the word "brand" really means rather than use expectation/perception because brand is not encompassed by either/both concepts. I am not talking about expectation and perception, I am talking about what something IS and how the IS drives the what something DOES. In this case, Wikipedia is the something. An IS is not necessarily visual. Brand is intangible but is expressed through that which is tangible, whether it be a mark/logo, or the way customer service responds to a customer or the way that a user experiences Wikipedia (not necessarily tangible but could be.) I have uploaded the graphic, if you need me to explain it, please ask.