I guess I'll throw my two cents in there..
Firstly, I've absolutely NEVER called it "philadelphia brand cream cheese", nor have I ever heard it.. Always just "philadelphia cream cheese" (you know, like the package actually says).. So, I'm not sure what made that such a great example, but it fell short..
Secondly, as stated, it seems you are conflating the terms "brand" and "name".. A name is simply a part of a brand, which is why you frequently hear the term "brand name".. The reason is because that is the name for that particular brand.. If the terms were interchangeable, people would just use the term "name" since it's the more common "everyday" term.. But they both hold value because they mean different things..
A brand is a perception or feeling, or sense of acknowledgment that surrounds a product.. It's a unifying set of expectations and consistencies commonly associated with a particular product.. I view it as a sort of ethereal or sublime intangible existence representative of a physical product.. An aspect of a brand is essentially a product's reputation..
If someone wants to change the reputation of their product, they need a new brand.. Not necessarily a new name, but a new identity.. The buccaneers are my quintessential example of this.. A franchise marred by futility completely rebranded and became immediate contenders, posting a record of .500 or better every year and making the playoffs in all but 1 season, leading up to a Super Bowl championship after only 6 years.. The teams logos, wordmarks, overall color scheme, visual appearance, uniform design, and even style of play and level of success all made a complete 180° at the same time.. That's about as "rebrand" as it gets..
Now, I'll also concede that there is a difference between rebrand and redesign, but I don't think size or magnitude really has as much to do with it as others have suggested.. The Miami Dolphins for example, have redesigned their uniforms.. They still feature an aqua dolphin on a coral (orange) sunburst, on a white helmet, with aqua jersey/white numbers/coral trim or white jersey/aqua numbers/coral trim, etc.. It's just a redesigned version of what already existed in several ways.. Aesthetically, I'd consider the dolphins a redesign (at least "on field", such as uniforms).. However, as people frequently point out that the logo evokes a cruise line or resort hotel feel, there are aspects of the organization that are clearly and markedly a "rebrand" (as confirmed by Brandon Moore, who made it clear that there was a big push to become a "luxury brand")..
Other examples to me would be the Seahawks, Vikings, and more recent Buccaneers redesigns.. All were strictly uniform-based, and simply put a "new spin" on existing logos, colours, and design elements, without changing much about the public's perception of these franchises..
However, I would argue that the Rams going back to royal/yellow, even if they keep the same basic uniform design, that it would be a rebrand.. They're moving to a brand that's more in line with the warm, sunny LA weather and bright, vibrant LA/Hollywood scenery.. That's a very deliberate and conscious brand decision.. It's also shifting back to a more marketable brand in terms of fan support and retail opportunities, with the popularity of the throwback merchandise and historic colors.. It's a clear shift in brand identity..
I don't suppose any of this will sway anyone one way or the other, but I thought I'd share my opinions on the matter, and further hammer the point home that "brand" does not mean "name" in current or former vernacular.. Perhaps its recent rise in popularity and usage stems from the fact that more and more people are recognizing it as the most correct and accurate term to describe these things..
That doesn't mean it can't be misused or misappropriated, but it also doesn't mean that each use you disagree with is a misuse or misappropriation..