No, not that another man was literally there, but that Bill (or the viewer) sees such an impression.
Compare the Dr. Strangelove cover which shows the round table in the War Room [1]. If you squint, you'll realize what you're looking at is actually a mushroom cloud. The table is the main part of the mushroom cloud, the men sitting around it are little explosions at ground level, and the circular light above the table is the ring around the mushroom cloud. This is obviously intentional, but there's no implication that a nuclear bomb literally exploded inside the war room.
Compare the Dr. Strangelove cover which shows the round table in the War Room [1]. If you squint, you'll realize what you're looking at is actually a mushroom cloud. The table is the main part of the mushroom cloud, the men sitting around it are little explosions at ground level, and the circular light above the table is the ring around the mushroom cloud. This is obviously intentional, but there's no implication that a nuclear bomb literally exploded inside the war room.
[1] https://img.goldposter.com/2015/04/Dr.-Strangelove-or-How-I-...