The name of the site is Stack Overflow. That name is stylized as stackoverflow in the logo because the person designing the picture included some text formatted a certain way. You can see similar treatments of logos for products, titles, and sites all over the world. For example:
- Heroku displays its name lowercase in its logo, but clearly uses an ordinary capitalized name when referring to itself in regular text.
- Similarly, Macy's renders its logo lowercase and even uses a star in place of the apostrophe.
- Nike takes things the other direction and writes its name in all caps next to its swoosh, but the name of the company is still just plain old Nike.
- Wikipedia's logo is in small caps, except that the final A is full-sized; this is for visual symmetry to match the initial W, not because the organization actually wants to be referred to as "WikipediA."
- Subway puts the first three letters of its name in one color, and the second three in another, but that doesn't mean they're to be interpreted as two separate words.
In the absence of other examples, use logos as guidance for how to write names of things, but remember that logos are still mainly display pieces, not mere text. People may think they're demonstrating how observant they are by mimicking a logo when typing a name, but they're often just revealing their ignorance in failing to look beyond the logo.
When people cite the trademark as the reason for editing, I think they're being a little overzealous. I don't need a trademark to tell me to capitalize proper nouns, or to recognize the difference between graphics that happen to include textual elements, and ordinary text.
The logo is fine the way it is. The different text weights suggest that they're two separate words, and our intuition tells us to capitalize them. Write it that way when you type the name, and correct people when they get it wrong, if you can do so politely. As more people learn and notice the distinction, the mistake will happen less often.