Exclusive Update: Erik Johnson, senior project leader at Valve, was kind enough to respond to my inquiry about the validity of the author's claim in the PC Gamer article. Erik confirmed to us that the plan is indeed to release Dota 2 this summer! Click the image if you want to see my initial email.
Original Post: Dota 2 will be released prior to The International 3, according to the July 2013 issue of PC Gamer.
Also, Reddit user exitwarp, who happens to be the author of the article, reiterated that Erik Johnson told him Dota 2 would be released this summer and it would be "free for everyone." Always keep in mind, though, that valve time is unpredictable. The goal to release the game this summer could stall if they decide that it needs a few more weeks or months of extra polish, but we're confident that this is going to be a done deal.
The International 3 will be held August 7 - 11 in Seattle.
This information and image credit goes to /u/Kakkoister.