Who decided to confront the Japanese at Midway?

In 2017, there was an actual release of the complete intelligent report (suppressed for 70 years) on Midway Island. The report was apparently the source for a newspaper article, which unwisely, was overly detailed in a Chicago Tribune story shortly after the Midway Island attack. To quote from the reference link:

While they were in route, Adm. Chester W. Nimitz, armed with the latest intelligence, "circulated a message to all of his commanders at sea giving them a little preview . . . about the battle of Midway that was going to occur in four or five days," Carlson said. Among other things, the crucial message - No. 311221 - laid out in detail the makeup of the enemy force.

At first, the United States was unsure where the enemy planned to attack. Japanese communications kept referring to a location code-named "AF." The Navy guessed it was Midway, but it had to be sure. To find out, Navy Com. Joseph J. Rochefort, a code breaker, suggested a ruse. Midway was instructed to issue an emergency call in plain English saying that its water distillation plant had broken down. The report was duly picked up by enemy eavesdroppers, who radioed superiors that "AF" was running short of water, according to Costello.

But how did the actual intelligence report get leaked? Here is an account:

The dispatch wound up in the hands of the Lexington's rescued executive officer, Cmdr. Morton T. Seligman, who happened to be bunking with Johnston. "So you put him in the same room with the dispatch, and the Navy and everybody else put two and two together. Much of the content of Nimitz's dispatch appeared in Johnston's story."

Now, as to why the Chicago Tribune continues to note its role in a questionable security leak, I am not sure. Yes, I am for freedom of the press, but why present such detail that any intelligence expert would be able to affirm its credibility on clearly a matter of immense national security? Seems like poor judgment to me.

The Trap set by the US Navy at Midway(June 4-7 1942) was Admiral Nimitz call. Although Commander in Chief, United States Fleet (COMINCH) Admiral Ernest King in Washington was in constant contact with Nimitz as he made this call. We know this because it was the Pacific Fleet's intelligence officer in charge of station HYPO, Joseph Rochefort under Nimitz, and his familiarity with the Japanese J-25b code who uncovered that Midway was the Japanese fleet's next target.

Office of Chief Of Naval Operations Signals Intelligence, OP-20-G based in Washington D.C. and station CAST then based in Australia; disagreed with Rocheford and Nimitz and thought the attack's target was probably the Aleutian Islands. possibly Port Moresby in Papua New Guinea, or even the west coast of the United States. Station HYPO then confirmed Midway was the target by requesting Midway to send an un-encrypted message about their desalination plant; which the Japanese intercepted and rebroadcast encoded. Thus verifying the identity of the disputed location AF was Midway.

Quote Given in T.E.D's answer
A persistent Midway myth is that the intelligence picture was so perfect, like reading the actual Japanese operation order, that even the day and time of the initial attack on Midway was known well beforehand. In truth, Nimitz worked only from a broad outline of Japanese intentions and partial order of battle, absolutely invaluable as they were.

The Midway intelligence was actually very specific and incredible accurate. The date and time of the attack, the degrees from Midway of the Japanese Carriers would approach from, and their distance from Midway was off by: "five minutes, five degrees, and five miles out". ( That's the difference between the intelligence officers statement before the battle began and the Japanese's Carrier's location after the initial attack on Midway ).

The Battle of Midway
Only an hour earlier Nimitz had asked Layton (his Intelligence officer) to give him a specific prediction of when and where the Japanese carriers would be first spotted. Layton swallowed hard and hazarded 0600, from the northwest at a bearing of 325 degrees, at a distance 175 miles from Midway. When Nimitz received the PBY's report in his operations room he could not resist tweaking his intelligence officer; turning to Layton he dryly commented, "Well, you were only five minutes, five degrees, and five miles out."

Codebreakers Set a Trap to Confirm Japanese Attack

By the end of May, Navy cryptanalysts had figured out more details about Yamamoto’s plans, including almost the entire order of battle of the Imperial Navy. With this information, Nimitz was able to plot a strategy that would take the Japanese by surprise, assembling three U.S. aircraft carriers at a spot some 300 miles north of Midway, which they called “Point Luck.”

Order of Battle
In modern use, the order of battle of an armed force participating in a military operation or campaign shows

• the hierarchical organization,
• command structure,
• strength,
• disposition of personnel,
• equipment of units and formations of the armed force.

I have in my backpack my shiny new xmas present, Shattered Sword, by Parshall and Tully. It mostly covers the Japanese side of things, and I'm barely into it, but the forward by John Lundstrom (who was the recipient of the first and most profuse of the authors' acknowledgements) credits the aggressive deployment decisions at both Coral Sea and Midway fully to Nimitz.

He also said that Nimitz felt he had good reasons to be optimistic about Midway:

To advance to Midway, Japan must expose its precious carriers to several forms of counterattack strongly abetted by the element of surprise. The U.S. carriers would finally enjoy a "fairly strong" land-based air umbrella to report and engage the enemy, whereas the Japanese carriers lacked similar support. An admiral who wore the dolphin insignia, Nimitz anticipated a stellar performance by his submarines.

… and some bad reasons …

Moreover, he, unlike the historians, did not realize the full weight of the resources arrayed against him. A persistent Midway myth is that the intelligence picture was so perfect, like reading the actual Japanese operation order, that even the day and time of the initial attack on Midway was known well beforehand. In truth, Nimitz worked only from a broad outline of Japanese intentions and partial order of battle, absolutely invaluable as they were.

Someone with access to Lundstrom's book, The First Team: Pacific Naval Air Combat from Pearl Harbor to Midway, may have more detail on the decision-making process on the American side, from which the above statements were drawn.