There is no difference between a nuclear bomb and an atomic bomb. The major types of explosive devices that operate by acting upon atomic nuclei are:
Pure Fission- Fat Man, Little Boy, Trinity all fall into this category. Compress the uranium until it reaches critical mass, and... KA-BOOM!
Thermonuclear- Modern ICBM warheads, they use a fission explosion to create a second, but much more powerful, fusion explosion
Neutron- A modified nuclear bomb that focuses on radiation release by using walls of chromium and nickel instead of lead
Dirty Bomb- An inside-out version of a pure fission device, it can be also thought of as a frag grenade with the outer shell being made of radioactive material. This is the 'WMD' that terrorists can most plausibly use because getting weapons-grade uranium or plutonium is no easy feat, but other radioactive materials are significantly more accessible.
Therefore, there is no difference between an atomic bomb and a nuclear bomb because they are both just different names for the same idea. A
thermonuclear device on the other hand, (which is what you may have been referring to)
is different from an 'atomic' weapon because it incorporates fusion, and I agree that Oppenheimer had not witnessed such an explosion. While that is an interesting point, it isn't relevant to the question of whether the US government had sufficient evidence at the time to make a decision that was sufficiently well-informed to justify the use of nuclear weapons.
You are aware that Trinity took place on the 16th of July, 1945, right?
Since you are obviously so full of yourself yet you are unable to comprehend the very words that I stated, and are again putting words in my mouth, I will say this:
The mention of hind sight was to show you that you have had decades of research, decisions, and theories presented to you to draw opinions on the actions of the leadership at the end of the war in the Pacific. Those same people who made the decisions did NOT have the luxury of time as you say, nor the vast amounts of information you possess. Further, another oft overlooked issue that was not previously mentioned to negate your belief in the allies having "time on their hands" was the issue of American POW's being held by the japanese on the home islands. Those personnel did not have time.
So to sum this up ONE LAST TIME, these facts are presented to support using nukes to force the japanese to surrender unconditionally:
- allied estimates of operation Olympic casualties were staggering (General LeMay estimated over half a million US dead, not casualties, dead. Herbert Hoover estimated between 500k and 1 million US dead. Secretary of War Stinson estimated between 5 - 10 million japanese dead.)
- the estimates of Kamikaze prepared aircraft alone were between 5 and 10 thousand aircraft. This does not include the multitude of other suicide weapons planned.
- japanese defense was expected to be fanatical based upon examples on Okinawa.
- continued bombing of japanese cities was more expensive in lives than the examples picked for nukes of two cities with four aircraft.
The last thing I am going to say to you with regards to the support of the decision to drop nukes is that had the US gone forward with Olympic and faced between 500k and 1 million US DEAD, how many would not be here today in the US? Further, how many japanese would not be here today? The luxury you have of second guessing those decisions is a gift, one that you should be thankful for as it is a strong possibility that many of us on these boards would not be here today had Olympic gone forward.
As for being full of myself, I'll quote the words of my robotics teacher: "It isn't being arrogant if you're right".

All jests aside, I don't think of myself as 'above' anyone. If you can prove your point, then it's right in my book. However, I still have some bones to pick.
Just because the had allies made an estimate doesn't prove that they had made a good estimate. I can prove it without hindsight, too. If the allied planners felt that they had good reason to believe that the Japanese would fight to the last even in the face of vast military superiority (e.g., Okinawa) then what difference would dropping a nuclear bomb on two cities have made to a foe that was clearly irrational? This presents an interesting paradox. If the Japanese were truly as irrational as you say, then the solution would have been to exterminate them. Obviously that wasn't on the allied agenda (they wanted to use the bombs to end the war), so even if it may have been true, the allied planners certainly didn't think so (which makes the Kyujo incident moot). That leaves the conclusion that Japan would have surrendered to a sufficiently superior military force.
As you have already demonstrated, the firebombing of Tokyo was insufficient to force a surrender; therefore, the shock and awe of a nuclear weapon clearly had some advantages. However, if the firebombing of Tokyo caused more casualties (albeit less quickly, but Japanese high command seeing their capital city burning would have clearly made up for it) than the bomb was predicted to, then it was not the damage part of shock and awe that interested the US. With that in mind, one can reason that isolating the magnitude of the explosion from the deaths of the denizens of Hiroshima and Nagasaki would have caused a negligible loss in shock and awe. That leaves the application of nuclear weapons to low-population areas such as Mount Fuji or other clearly recognizable landmarks that would have sufficient eye-witnesses (to prevent a cover-up) coupled with a very threatening letter, as the best possible way to end the war.
With regard to the POW's, the casualties generated by leaving them in there for the few days necessary would have been negligible compared to the number of casualties generated by nuking Hiroshima and Nagasaki.
-Penguin