People should be addressed as they prefer to be addressed. If someone says her name is Elizabeth, call her Elizabeth, not Liz or Beth or Betsy. (Ignore her if she insists on being addressed as Your Royal Highness, unless she lives in Windsor Castle.)
Likewise, we should call companies by the names they have chosen for themselves. Even though I might prefer the name Hewlett-Packard, the company officially changed its name to HP in 2015. Apple hasn’t been Apple Computer since 2007.
That doesn’t mean, however, that we have to reproduce the company logo when we write about them.
At first, that sounds like a no-brainer. We couldn’t reproduce most logos if we wanted to. Good luck putting the E in Dell at a 45-degree angle, or putting arrows on the S and Y in Subway.
But wait: If I look on Subway’s website, the sandwich company always writes its own name as SUBWAY. So shouldn’t we do the same? Nope. All those capital letters are just an attempt to approximate the logo. The company gets to pick its own name, but we are still free to treat it like any other proper name.
So it’s Subway and Starbucks, even though their logos are in all capital letters. Likewise, it’s Amazon and Intel, even though their logos are in all lower-case letters. It’s HP, not hp.
And even though we have a lot of three-letter TV networks such as ABC, CBS, NBC and PBS, the outfit run by Rupert Murdoch is Fox, not FOX. It should be easy to see why. Those other network examples are initialisms. Fox Broadcasting and Fox News (two separate divisions of 21st Century Fox) get their names from William Fox, who founded Fox Film Corp. in 1915.
And yes, the parent company calls itself 21st Century Fox these days. It still has a movie subsidiary called 20th Century Fox. (I never said this would be simple.)
Readers expect to see proper names with a capital letter at the start and lower-case letters after that. Sometimes we’ll bend the rules, such as for eBay, but only to make it easy for the readers to understand. Give priority to them, not to corporate marketing departments.