What name/term did high-level Manhattan Project participants use to refer to the project contemporaneously?

I am curious what people working on the Manhattan Project or coordinating with the project, called it in verbal discussions, when it was going on. (Specifically those high enough to be authorized to know that it was intended to create an atomic bomb, so my guess is that there are maybe 100-1000 people that would have fell into this category.)

Also, when did the term "Manhattan Project" (as opposed to "Manhattan District" or "Manhattan Engineer District") first come into widespread use?

I have done a fair bit of research so far but am not a professional historian; from my research so far:

• the Quebec Agreement called it Tube Alloys, presumably in deference to the British (although Oak Ridge chemists were instructed to use "tuballoy" rather than "uranium")

• DOE history (https://www.osti.gov/opennet/manhattan-project-history/People/MilitaryOrgs/army-corps.html) refers to "Department of Substitute Materials" which was quickly superseded by "Manhattan District":

  Engineer districts usually took their names from the city where they were located, but with Marshall's district lacking a permanent headquarters the cover name Laboratory for the Development of Substitute Materials, or DSM, initially was used. To Groves, however, the term DSM seemed likely to arouse attention and curiosity, and he and Marshall agreed that the name Manhattan would be a better name. On August 13, General Reybold issued a general order officially establishing "a new engineer district, without territorial limits, to be known as the Manhattan District."

• Memorandum on April 23 1945 from Groves to Secretary of War Stimson:

  The entire program is known as the "Manhattan District Project" and at times either the "DSM Project" or the "Tubealloy Project"

• Various correspondence between the White House and Bush or Conant often referred obliquely (e.g. "in your conversation yesterday" or "the special project") or to the S-1 Project (makes sense since they were part of the S-1 Executive Committee on Uranium)

• James Byrnes, as Director of the Office of War Mobilization, composed a memorandum to the President on Mar 2 1945, in which he refers to "the Manhattan project", and sent a copy to the President the next day --- this is the first written usage I can find:

  I understand that the expenditures for the Manhattan project are approaching 2 billion dollars with no definite assurance yet of production.

Also, although there were names for specific divisions of the project (Los Alamos was "Project Y", there was "Hanford Engineer Works" in Washington and "Clinton Engineer Works" in Tennessee with the various plants like Y-12, K-25, S-10, X-10) and the Los Alamos scientists used the term "gadget" to refer to various individual designs / configurations of fission weapons, like the one used at Trinity Site or the Hiroshima/Nagasaki bombs --- but what term would, say, the scientists or Hanford/Oak Ridge plant supervisors use when they were in a secure location and able to have a frank discussion in which they needed to refer to the project as a whole?