Mostly correct, but not entirely.
The first versions of DoTA in War3 Reign of Chaos (Vanilla war3) were created by Eul. The original Anti-mage (mageslayer / demon hunter) was named Eul, not this "Magina" crap that he is now. I used to play TONS of old school DoTA back in the day.
When War3: Frozen Throne came out, that's when Guinsoo developed DoTA: All Stars. I'm not entirely sure about this though, I stopped playing DoTA when FT came out and didn't start again until late 2006. I missed an entire 2 years of it's development.
Later on, with version 6++ Icefrog picked up Dota.
Also, I think saying that "Riot Games" is trying to copyright it is kind of misleading. As I understand, it's Guinsoo and Pendragon that are trying to copy right it, not Riot Games itself, even though they now work for Riot.
Here are my personal opinions on it. I hope that the name DoTA remains public domain as a mod. My personal hopes are that no *one specifically profits* from the name, ESPECIALLY not a corporation like Valve or Riot Games, even though the three developers Eul / Guinsoo / IceFrog have put in tons and tons of their personal time and now work for these companies.
To give a bit more insight on the situation:
DotA was created and maintained by a guy named Guinsoo for the first few years of it's life and Pendragon ran the most popular DotA community website. But both of them publicly gave up on the project aroun 2005 to go create Riot Games and develop their popular title League of Legends. IceFrog picked up the development of DotA after and continued updating it until Valve hired him.
So now Valve has filed for the trademark on the name DotA. It's a big deal since most games in the genre call themselves "Dota-likes" or just call the genre dota in general (instead of the more obscure MOBA classification). Of course, Riot Games and other DotA-like game developers do not want to be locked out of that designation. So Riot Games is counter-filing and state that if they win the trademark that the name of DotA will be free for all to use. But it's going to be a case of who has the stronger claim on the name.
Guinsoo - Original developer of DotA, but publically renounced his part on the project in 2005 which may be what seals this case shut. Had he never made a public forums post about it his case would be much more solid.
IceFrog - Picked up and maintained DotA after Guinsoo left. DotA arguably went through it's greatest expansion in popularity during his time in developing it.
But US trademark/patent law is incredibly complex. I doubt Valve has any real interest in potentially alienating customers and maybe this will get settled out of court.