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"I mentioned it to you shortly after you took office ..."
Yes that he needed to speak about something important. Draw your own implication but it doesn't actually say what your implying it says.
And if you wanted to know what Truman thought of not being told as vice president. (from the same link)
On June 17, Truman received a phone call from Stimson, who told him that the Pasco plant was “part of a very important secret development.” Fortunately, Stimson did not need to explain further: Truman, a veteran and a patriot, understood immediately that he was treading on dangerous ground. Before Stimson could continue, Truman assured the secretary “you won t have to say another word to me. Whenever you say that [something is highly secret] to me that’s all I want to hear. If [the plant] is for a specific purpose and you think it’s all right, that’s all I need to know.” Stimson replied that the purpose was not only secret, but “unique.”
Yes there was a very ambiguous statement about a secret project, that's not the same as being read in.
What he was told about could have been a large conventional bomb like the grand slam, tallboy, or firebats which were also secret at the time, there's a pretty big difference between them. I imagine there was probably quite a bit of hyperbole from secret project directors just as there is today so that's not a super great metric to judge things on.
You failed to quote the July 25th 1945 entry when he wrote about what he was actually told and shown which is as far as I'm aware the first definitive proof he knew exactly what it was.
You failed to quote the July 25th 1945 entry
That's Truman's reaction to the trinity test and is irrelevant to when he was told about the research. Obviously he had a different reaction to the actual bomb compared to a hypothetical bomb, but I would think that would be expected.
Yes the first mention of atomics and the first verified point he knew the full scale of the program, I just said this.