I'm not sure whether any of the 808nm "IR" (which is very faintly visible as laser radiation) makes it out of a green laser as it's not emitted along the axis afaik, or whether only the 1064nm component (which is definitely invisible) does...
For the BR, the solution might be that the near-UV causes many substances to fluoresce. In my experience, the most common resultant colors are green and a pink/red. Quite possibly the dyes fluoresce a bit as well. This might also be the explanation why you see red with a green laser.
This would also explain why you can see the red only by looking through your goggles: then the light is focused by your eyes. Since it's not laser radiation, it gets emitted in all directions and can't project a dot.